Class 10 Physics Chapter 1 Force

Key Concept: Translational Motion

b) Pushing a ball lying on a floor.
[Solution Description]
Translational motion occurs when a force acts on a stationary rigid body free to move, causing it to move in a straight path in the direction of the force.
Among the given options:
- Pushing a ball lying on a floor causes it to roll in a straight line, demonstrating translational motion.
- Other options involve rotational motion or no motion at all.

Key Concept: Force, Moment of Force, Rotational Motion

b) 30 N m
[Solution Description]
The moment of force (torque) is calculated using $\tau = F \times d$, where $F$ is the force and $d$ is the perpendicular distance from the pivot. Here, $F = 20 \text{ N}$, $d = 1.5 \text{ m}$. Therefore, $\tau = 20 \times 1.5 = 30 \text{ N m}$.

Key Concept: Rotational Motion

c) 10 N m
[Solution Description]
Torque ($\tau$) is calculated using the formula $\tau = F \times d$, where $F$ is the force and $d$ is the perpendicular distance from the pivot (radius in this case).
Given: $F = 5 \text{ N}$, $d = 2 \text{ m}$.
$\tau = 5 \times 2 = 10 \text{ N m}$.
The torque produced is $10 \text{ N m}$.

Key Concept: Moment of Force, Rotational Motion

b) 5 N m
[Solution Description]
The moment of force (torque) is calculated using the formula:
$\text{Torque} = F \times d$
where $F$ is the force and $d$ is the perpendicular distance from the pivot.
Substituting the given values:
$\text{Torque} = 10 \, \text{N} \times 0.5 \, \text{m} = 5 \, \text{N m}$
Since the rod is uniform and pivoted at its center, there is no other torque acting in the opposite direction.

Key Concept: Moment of Force

b) 20 N m
[Solution Description]
Moment of force (torque) is calculated using the formula:
$\text{Torque} = \text{Force} \times \text{Perpendicular distance}$
Given:
Force = 20 N,
Distance = 1 m
Therefore,
$\text{Torque} = 20 \text{ N} \times 1 \text{ m} = 20 \text{ N m}$

Key Concept: Translational and Rotational Motion

b) The body must be pivoted at a point
[Solution Description]
For rotational motion, the body must be pivoted at a point, and the force should be applied at a suitable point such that it creates a turning effect (torque). The body cannot be free to move entirely; otherwise, it would result in translational motion.

Key Concept: Translational Motion

c) The body moves in a straight path in the direction of the force
[Solution Description]
When a force acts on a stationary rigid body that is free to move, the body starts moving in a straight path in the direction of the force. This motion is called translational or linear motion.

Key Concept: Principle of Moments

b) 20 N
[Solution Description]
According to the Principle of Moments, for equilibrium:
$\text{Sum of clockwise moments} = \text{Sum of anticlockwise moments}$
Let $W$ be the unknown weight. The clockwise moment is $30 \, \text{N} \times 2 \, \text{m} = 60 \, \text{N m}$. The anticlockwise moment is $W \times 3 \, \text{m}$. For equilibrium:
$W \times 3 = 60$
$W = \frac{60}{3} = 20 \, \text{N}$

Key Concept: Moment of Force

d) Newton-metre (N m)
[Solution Description]
The moment of force (or torque) is calculated as $\text{Moment of force} = \text{Force} \times \text{Perpendicular distance}$. Its SI unit is newton-metre ($N m$).

Key Concept: Units conversion, moment of force

c) 5 kgf$\cdot$m
[Solution Description]
Conversion steps:
1. Given relation: $1\, kgf\cdot m = 9.8\, N\cdot m$
2. Therefore, $49\, N\cdot m = \frac{49}{9.8} = 5\, kgf\cdot m$

Key Concept: Clockwise and Anticlockwise Moments

d) Negative clockwise moment
[Solution Description]
According to the syllabus, if a force causes a body to rotate clockwise, the moment of force is called the clockwise moment and is taken as negative. Conversely, an anticlockwise moment is taken as positive.

Key Concept: Moment of Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The moment of force (torque) is given by $\text{Moment} = F \times d_{\perp}$, where $F$ is the force and $d_{\perp}$ is the perpendicular distance from the axis of rotation to the line of action of the force. When the force is applied perpendicular to the line joining the point of application and the axis, $d_{\perp}$ is maximum because the entire distance contributes to the moment. Thus, the Assertion is true. The Reason states that the perpendicular distance determines the magnitude of the moment, which is also correct because the formula explicitly depends on $d_{\perp}$. Moreover, the Reason correctly explains why the Assertion is true since maximizing $d_{\perp}$ maximizes the moment.

Key Concept: Translational Motion

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that applying a force to a rigid body free to move causes it to move in the direction of the force. This is correct as per the syllabus statement: "When a force acts on a stationary rigid body which is free to move, the body starts moving in a straight path in the direction of force." The reason claims that linear motion occurs only when the net force is non-zero. While this aligns with Newton's First Law ($\sum F = 0$ implies no acceleration), it does not explain why the motion must be in the direction of the applied force. Thus, both statements are true, but the reason does not fully explain the assertion.

Key Concept: Relation to Force and Motion, Units of Force

b) 19.6 m/s$^2$
[Solution Description]
First, convert the force from kgf to N:
$10 \text{ kgf} = 10 \times 9.8 \text{ N} = 98 \text{ N}$
Using Newton's second law, $F = ma$, we can find the acceleration ($a$):
$a = \frac{F}{m} = \frac{98}{5} = 19.6 \text{ m/s}^2$
Thus, the acceleration of the block is 19.6 m/s$^2$.

Key Concept: Force and Linear Motion

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a book does not move when pushed gently due to the balance between the applied force and friction. This is true, as static friction prevents motion until the applied force exceeds the maximum static friction. The reason claims that translational motion requires a non-zero net force, which is correct based on Newton's first law of motion. Since the reason correctly explains why the book does not move (net force is zero), both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Torque Calculation

c) 4.5 N m
[Solution Description]
The torque (\tau) is calculated using the formula:
$\tau = F \times d$
where:
$F = 15 \, \text{N}$,
$d = 0.3 \, \text{m}$.
$\tau = 15 \times 0.3 = 4.5 \, \text{N m}$
The torque produced is 4.5 N m.

Key Concept: Moment of force

b) 10 N m
[Solution Description]
The moment of force (torque) is calculated using the formula:
$\text{Moment of force} = \text{Force} \times \text{Perpendicular distance from the pivot}$
Substituting the given values,
$\text{Moment of force} = 20\, \text{N} \times 0.5\, \text{m} = 10\, \text{N m}$

Key Concept: Principle of Moments

a) 0 N m
[Solution Description]
The resultant moment is the sum of individual moments. Since the forces act in opposite directions, one creates a clockwise moment and the other an anticlockwise moment. Each has a lever arm of 2 m (half of 4 m).
$\text{Moment}_1 = 10\, \text{N} \times 2\, \text{m} = 20\, \text{N m (clockwise)}$
$\text{Moment}_2 = 10\, \text{N} \times 2\, \text{m} = 20\, \text{N m (anticlockwise)}$
The resultant moment is:
$20\, \text{N m} - 20\, \text{N m} = 0\, \text{N m}$

Key Concept: Clockwise and Anticlockwise Moments

c) Applying force at the point farthest from the pivot
[Solution Description]
According to the syllabus, the maximum turning effect is produced when the perpendicular distance of the line of action of the force from the axis of rotation is maximum.

Key Concept: Moment of Force, Couple

b) 10 Nm clockwise
[Solution Description]
The net moment is calculated by summing the clockwise and anticlockwise moments.
- Moment due to force at 0.5 m: Since it is on one side of the pivot, let's assume it creates an anticlockwise moment (positive).
$\tau_1 = F \times d = 10 \, \text{N} \times 0.5 \, \text{m} = +5 \, \text{Nm}$.
- Moment due to force at 1.5 m: Applied in the opposite direction, so it creates a clockwise moment (negative).
$\tau_2 = -F \times d = -10 \, \text{N} \times 1.5 \, \text{m} = -15 \, \text{Nm}$.
Net moment: $\tau_{\text{net}} = \tau_1 + \tau_2 = 5 \, \text{Nm} - 15 \, \text{Nm} = -10 \, \text{Nm}$.
Thus, the net moment is 10 Nm in the clockwise direction.

Key Concept: Moment of Couple, Torque

b) 15 Nm
[Solution Description]
A couple consists of two equal and opposite forces acting at a distance apart.
The moment of the couple is given by:
$\tau = F \times \text{Couple arm}$.
Here, the couple arm is the diameter of the wheel:
$r = 2 \times 0.3 \, \text{m} = 0.6 \, \text{m}$.
Thus, $\tau = 25 \, \text{N} \times 0.6 \, \text{m} = 15 \, \text{Nm}$.
The direction of the moment depends on the applied forces but the magnitude is 15 Nm.

Key Concept: Direction of Moment, Net Moment

b) 2.5 N m anticlockwise
[Solution Description]
Calculate the moment due to the 15 N force (clockwise):
$M_1 = 15 \, \text{N} \times 0.5 \, \text{m} = 7.5 \, \text{N m} \, (\text{clockwise})$
Calculate the moment due to the 10 N force (anticlockwise):
$M_2 = 10 \, \text{N} \times 1 \, \text{m} = 10 \, \text{N m} \, (\text{anticlockwise})$
The net moment is the difference between the two moments:
$M_{\text{net}} = M_2 - M_1 = 10 \, \text{N m} - 7.5 \, \text{N m} = 2.5 \, \text{N m} \, (\text{anticlockwise})$

Key Concept: Moment of Force, Perpendicular Distance

b) 7.071 N m
[Solution Description]
To find the effective moment, we need the perpendicular component of the force relative to the wrench handle:
$F_{\perp} = F \times \sin(\theta) = 50 \, \text{N} \times \sin(45^\circ) = 50 \times 0.7071 = 35.355 \, \text{N}$
Now, calculate the moment:
$M = F_{\perp} \times d = 35.355 \, \text{N} \times 0.2 \, \text{m} = 7.071 \, \text{N m}$

Key Concept: Balancing moments

c) 30 cm
[Solution Description]
Let the distance from the pivot for the 10 N weight be $x$. To balance the moments: Clockwise moment = Anticlockwise moment. The 5 N weight creates a clockwise moment: $5 \, \text{N} \times (60 \, \text{cm} - 40 \, \text{cm}) = 5 \, \text{N} \times 20 \, \text{cm} = 100 \, \text{N\,cm}$. The 10 N weight must create an equal anticlockwise moment: $10 \, \text{N} \times (40 \, \text{cm} - x) = 100 \, \text{N\,cm}$. Solving for $x$: $40 \, \text{cm} - x = 10 \, \text{cm}$, so $x = 30 \, \text{cm}$. Thus, the 10 N weight should be placed at the 30 cm mark.

Key Concept: Principle of Moments

c) 1 m
[Solution Description]
For equilibrium, the moments on both sides must be equal.
Let the distance of the second child be $d$.
The anticlockwise moment is $40 \, \text{kg} \times 1.5 \, \text{m} = 60 \, \text{kg m}$.
The clockwise moment must also be 60 kg m, so $60 \, \text{kg} \times d = 60 \, \text{kg m}$.
Solving gives $d = 1 \, \text{m}$.

Key Concept: Principle of Moments, Multiple Forces

b) 47.5 cm
[Solution Description]
Calculate the moments due to existing forces:
$\text{Anticlockwise moment} = 10\,N \times (40\,cm - 10\,cm) = 10\,N \times 30\,cm = 300\,N\,cm$
$\text{Clockwise moment} = 5\,N \times (70\,cm - 40\,cm) = 5\,N \times 30\,cm = 150\,N\,cm$
Net unbalanced moment is $300\,N\,cm - 150\,N\,cm = 150\,N\,cm$ anticlockwise. To balance this, the 20 N weight must provide an equal clockwise moment. Let its distance from the pivot be $x$:
$20\,N \times x = 150\,N\,cm \implies x = \frac{150}{20}\,cm = 7.5\,cm$
Since it must be on the opposite side, its position is $40\,cm + 7.5\,cm = 47.5\,cm$.

Key Concept: Principle of Moments

b) Sum of clockwise moments equals sum of anticlockwise moments
[Solution Description]
For equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments. Here, assume the heavier child (60 kg) is on one side and the lighter (40 kg) on the other. The moment due to the 60 kg child is $60 \times 1.5 = 90 \, \text{kgf m}$ clockwise. The moment due to the 40 kg child is $40 \times 2 = 80 \, \text{kgf m}$ anticlockwise. However, for equilibrium, these must balance, which they do not. Hence, the correct condition is when both produce equal moments by adjusting distances. The correct answer is when sums are equal, as per the principle of moments.

Key Concept: Couple

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A couple consists of two equal and opposite parallel forces that do not act along the same line, resulting in a net force of zero. This means there is no translational motion. However, since the two forces create moments in the same direction, they produce a rotational effect. Therefore, the Assertion is true. The Reason correctly explains why the Assertion is true because the zero resultant force indeed leads to no translational motion, while the moment causes rotation. Thus, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.

Key Concept: Direction of Rotation

b) Rotation without translation
[Solution Description]
When two equal and opposite parallel forces act on a body with different lines of action, they form a couple. A couple produces pure rotational motion without any translational motion because the net force is zero but there is a net turning effect.
The correct answer is rotation without translation.

Key Concept: Couple, Moment of Couple

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To evaluate the assertion and reason:
1. **Assertion Analysis**: The moment of a couple is indeed independent of the point about which it is calculated because it depends solely on the product of either force and the perpendicular distance between them (couple arm). This is a fundamental property of couples.
2. **Reason Analysis**: The reason correctly states that a couple consists of two equal and opposite parallel forces with non-coinciding lines of action. This is the definition of a couple.
3. **Connection**: While the assertion is true and the reason is also true, the reason does not explain why the moment is independent of the point. It is a property derived from the definition but not directly explained by it. Thus, the reason is not the correct explanation of the assertion.
Hence, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Moment of Couple Definition

c) Two equal and opposite parallel forces with different lines of action
[Solution Description]
A couple is formed by two equal and opposite parallel forces that are not acting along the same line. This pair of forces produces a rotational effect without any translational motion. The moment of a couple is calculated as the product of one of the forces and the perpendicular distance between them, given by $F \times d$.

Key Concept: Moment of couple, rotational equilibrium

c) 22 Nm
[Solution Description]
The net moment is the sum of the moments of the two couples. First couple: $M_1 = 5 \times 2 = 10$ Nm. Second couple: $M_2 = 3 \times 4 = 12$ Nm. Since both couples act in the same direction, the net moment is $M_1 + M_2 = 10 + 12 = 22$ Nm.

Key Concept: Moment of Couple, Perpendicular Distance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The moment of a couple is given by $M = F \times d$, where $F$ is the magnitude of either force and $d$ is the perpendicular distance between the two forces. This means it depends only on these two quantities. The reason states that the moment of a couple is independent of the point about which it is calculated because the resultant force is zero. This is true since a couple does not produce any translational motion and its rotational effect (moment) remains the same regardless of the reference point. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Centre of Gravity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The centre of gravity of a triangular lamina is indeed at the point of intersection of its medians. This is because the mass distribution is uniform in a lamina, and the medians divide it into smaller triangles of equal area (and thus equal weight). The reason correctly states that the centre of gravity is the point where the entire weight of the body can be considered to act. Both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Centre of Gravity for Irregular Lamina

d) To accurately locate the single centre of gravity by intersecting plumb lines
[Solution Description]
Suspending the lamina from different points helps trace multiple vertical lines (plumb lines) on the lamina. The intersection point of these lines gives the centre of gravity because, in each case, the C.G. must lie vertically below the point of suspension. Using multiple points ensures accuracy in determining the exact location of the C.G.

Key Concept: Centre of Gravity Position

d) At the intersection of its diagonals
[Solution Description]
For a uniform rectangular lamina, the centre of gravity lies at the intersection of its diagonals.

Key Concept: Centre of Gravity

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that the centre of gravity of a uniform triangular lamina lies at the intersection of its medians. This is true because, for a uniform triangular lamina, the mass distribution is symmetrical about the medians, and their intersection point satisfies the condition $\sum \vec{r}_i \times \vec{w}_i = 0$.
The Reason claims that for any uniform two-dimensional shape, the centre of gravity coincides with its geometric centre. While this is true for symmetric shapes like circles or rectangles, it does not universally hold for all two-dimensional shapes (e.g., irregular laminae may have their C.G. elsewhere). Thus, the Reason provides an oversimplified explanation and does not correctly explain the Assertion.

Key Concept: Centre of Gravity Experiment

c) By suspending it from two different points and intersecting the plumb lines.
[Solution Description]
To determine the centre of gravity of an irregular lamina, suspend it freely from a point and allow it to come to rest. Draw a vertical line (plumb line) below the suspension point. Repeat this process by suspending the lamina from another point and drawing another plumb line. The intersection point of these two lines gives the centre of gravity. If more accuracy is needed, repeat the process with a third suspension point to confirm the location.

Key Concept: Position of Centre of Gravity

c) At the mid-point of the rod
[Solution Description]
For a uniform rod, the mass distribution is symmetric and equal on both sides. Therefore, the centre of gravity lies at its midpoint.

Key Concept: C.G. Outside Material

c) Hollow sphere.
[Solution Description]

A hollow sphere has its centre of gravity at its geometric centre, which is a point not occupied by any material of the sphere.

Key Concept: Definition of Centre of Gravity

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that for a uniform rectangular lamina, the C.G. lies at the intersection of its diagonals. This is true as per the syllabus examples. The reason states that for regular shaped objects, the C.G. coincides with the geometrical centre. This is also true as per the syllabus. However, the reason does not specifically explain why the C.G. of a rectangular lamina is at the intersection of diagonals - it just gives a general property. Therefore, both are true but reason is not the correct explanation of assertion.

Key Concept: Centre of Gravity, Deformation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The position of the centre of gravity depends on the distribution of mass in the body. For a uniform rod, the centre of gravity is at its midpoint. When the rod is bent into a semi-circular shape, the mass is redistributed along the arc of the semicircle. This redistribution causes the centre of gravity to shift from the original midpoint to a new position along the vertical axis of symmetry of the semicircle.
The reason correctly explains why the assertion is true because altering the shape of the body directly affects how mass is distributed, leading to a change in the position of the centre of gravity.

Key Concept: C.G. of solid cone vs hollow cone

b) Solid cone's C.G. is higher than hollow cone
[Solution Description]
For a solid cone, C.G. is at $\frac{2h}{3}$ from vertex while for hollow cone it's at $\frac{h}{3}$ from vertex. This is because in hollow cone, more mass is concentrated near base making C.G. lower compared to solid cone where mass is evenly distributed.

Key Concept: Centre of Gravity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a uniform circular ring, the mass is distributed symmetrically around its geometric centre. Since the distribution of mass is uniform and symmetric, the algebraic sum of moments due to all particles about the centre will be zero. Hence, the C.G. of a uniform circular ring lies at its geometric centre. The reason correctly explains the assertion because symmetry ensures equal distribution of mass around the centre, making it the C.G.

Key Concept: Centre of Gravity for Regular-Shaped Objects

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a circular disc, the mass is uniformly distributed, which means every part of the disc has equal mass per unit area. Due to this symmetry, the centre of gravity coincides with the geometric centre of the disc. Hence, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Centre of Gravity Determination

c) To minimize errors and confirm the intersection point as the true centre of gravity
[Solution Description]
Suspending the lamina from only one or two points may not give accurate results because the intersection point of lines drawn from fewer suspension points might not accurately represent the true centre of gravity. Using three points ensures that any error in alignment or measurement is minimized, and the intersection point confirms the position of the centre of gravity.

Key Concept: Irregular lamina, Plumb line accuracy

b) Higher number of points increases accuracy by confirming consistency
[Solution Description]
The accuracy of locating the center of gravity (G) depends on:
1. Multiple suspension points help verify consistency in the intersection of plumb lines.
2. More suspension points reduce errors by averaging out discrepancies in individual measurements.
3. Three points are typically sufficient to confirm convergence, but additional points improve reliability.
Thus, increasing the number of suspension points enhances the confidence in pinpointing the true center of gravity.

Key Concept: Experimental determination of C.G. using a plumb line

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion describes the experimental method used to locate the centre of gravity (C.G.) of an irregular lamina by suspending it from different points and marking the plumb lines. The reason explains why this method works: when a body is freely suspended, it balances in such a way that its C.G. lies directly below the suspension point. Thus, the plumb line method relies on this principle for accurate determination of C.G.

Key Concept: Centre of Gravity

c) At the intersection of its diagonals.
[Solution Description]
The centre of gravity of a uniform body coincides with its geometric centre because the mass distribution is symmetrical. For a uniform rectangular block, the centre of gravity is at the intersection of its diagonals.

Key Concept: Principle of Moments

a) The sum of the clockwise moments equals the sum of the anticlockwise moments.
[Solution Description]
For the metre rule to be in equilibrium, the algebraic sum of moments about the pivot must be zero. The moment due to the 5 N weight (clockwise) is $5 \times 30 = 150 \text{ N cm}$. The moment due to the 10 N weight (anticlockwise) is $10 \times 20 = 200 \text{ N cm}$. Since $150 \neq 200$, the rule is not in equilibrium unless an additional moment balances the difference.

Key Concept: Static Equilibrium

b) A book lying on a table
[Solution Description]
Static equilibrium occurs when a body remains at rest under the influence of balanced forces. In the given examples, a book lying on a table shows static equilibrium because the downward force (weight of the book) is balanced by the upward normal force exerted by the table.

Key Concept: Dynamic Equilibrium, Centripetal Force

d) 8000 N
[Solution Description]
In dynamic equilibrium, the net force provides the centripetal force required for circular motion.
Centripetal force formula: $F_c = \dfrac{m v^2}{r}$.
Given:
Mass $m = 1000$ kg,
Speed $v = 20$ m/s,
Radius $r = 50$ m.
Substituting values:
$F_c = \dfrac{1000 \times (20)^2}{50} = \dfrac{1000 \times 400}{50} = 8000$ N.
Since the car is in dynamic equilibrium, this is also the net force acting on it.

Key Concept: Static Equilibrium

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Static equilibrium occurs when a body remains at rest under the influence of balanced forces. In this case, the book's weight ($W$) acts downward due to gravity, and the table exerts an equal and opposite normal reaction force ($N$) upward. Since these two forces are equal and opposite, their sum is zero ($\sum F = 0$). Therefore, the book does not accelerate and remains in static equilibrium. The reason correctly explains why the assertion is true.

Key Concept: Principle of Moments

b) Sum of clockwise moments equals sum of anticlockwise moments
[Solution Description]
The principle of moments states that for a body to be in rotational equilibrium, the sum of clockwise moments about any point must equal the sum of anticlockwise moments about the same point. Mathematically represented as $\sum$(clockwise moments) = $\sum$(anticlockwise moments).

Key Concept: Static Equilibrium

b) Static equilibrium
[Solution Description]
The book remains at rest on the table because the downward force (weight) is balanced by the upward normal force from the table. This satisfies the condition for static equilibrium where the net force is zero and there is no motion.

Key Concept: Static Equilibrium

d) The weight of the book and the normal reaction are equal and opposite.
[Solution Description]
The book is in static equilibrium when the net force acting on it is zero. The weight of the book acts vertically downward, and the normal reaction from the table acts vertically upward. These forces are equal and opposite, balancing each other.

Key Concept: Conditions for Equilibrium

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for a body to be in equilibrium, the resultant of all forces acting on it must be zero. This is correct as per the first condition of equilibrium \[\sum F = 0\]. The reason given is that this ensures no translational acceleration of the body, which is also correct and directly explains why the resultant force must be zero for equilibrium. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Static Equilibrium, Moment Calculation

b) $784\, \text{N}$
[Solution Description]
Let $N_w$ be the normal force from the wall, and $N_f$ the normal force from the floor.
Step 1: Resolve forces vertically ($\sum F_y = 0$):
$N_f = mg = 80\, \text{kg} \times 9.8\, \text{m/s}^2 = 784\, \text{N}$
Step 2: Take moments about the base ($\sum \tau = 0$):
The weight acts at $2.5\, \text{m} \cos 30^\circ$ from the base.
$N_w \times 5\, \text{m} \sin 30^\circ = 784\, \text{N} \times 2.5\, \text{m} \cos 30^\circ$
Solve for $N_w$:
$N_w = \frac{784 \times 2.5 \cos 30^\circ}{5 \sin 30^\circ}$
Simplify using $\sin 30^\circ = 0.5$ and $\cos 30^\circ = \sqrt{3}/2$:
$N_w = \frac{784 \times 2.5 \times \sqrt{3}/2}{5 \times 0.5} = 196\sqrt{3}\, \text{N}$
However, the question asks for $N_f$, which we calculated as $784\, \text{N}$.

Key Concept: Dynamic Equilibrium

d) The sum of buoyant force and drag force equals the gravitational force.
[Solution Description]
In dynamic equilibrium, the net force acting on the object is zero because it moves with constant velocity. Here, the downward gravitational force (weight) is balanced by the upward buoyant force and air resistance (drag force). Thus, these forces cancel each other out, resulting in no acceleration.

Key Concept: Dynamic Equilibrium, Net Force Balance

c) $v_t = \sqrt{\frac{mg}{k}}$
[Solution Description]
In dynamic equilibrium, net force is zero ($\sum F = 0$):
$mg = F_d$
Given $F_d = kv_t^2$, substitute:
$mg = kv_t^2$
Solve for $v_t$:
$v_t = \sqrt{\frac{mg}{k}}$

Key Concept: Couple, Moment of force

b) 5 Nm initially, 50% increase
[Solution Description]
Moment of couple = Force × perpendicular distance between forces.
Here, distance = spanner length = 20 cm = 0.2 m.
Initial moment = $25\,N \times 0.2\,m = 5\,Nm$.
When spanner length increases to 30 cm (0.3 m), new moment = $25 \times 0.3 = 7.5\,Nm$.
Percentage increase = $\frac{7.5-5}{5} \times 100 = 50\%$.

Key Concept: Principle of Moments

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a door is easier to open when the force is applied at the handle rather than near the hinges, which is true because the handle is placed farther from the pivot (hinges), increasing the moment arm. The reason explains that the moment of force increases with an increase in the perpendicular distance from the pivot point, which is also true and directly explains why applying force at the handle makes it easier to open the door. Hence, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Moment of Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that opening a door becomes easier when the force is applied at the farthest point from the hinges, which is true because the moment of force ($\text{Moment} = \text{Force} \times \text{Perpendicular distance}$) increases with distance. The reason correctly explains this phenomenon since a greater perpendicular distance results in a larger turning effect (moment), making it easier to open the door. Thus, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Equilibrium, Centre of gravity

b) 3.06 kg
[Solution Description]
For equilibrium, sum of moments about P must be zero. Let $m$ be mass in kg.
Clockwise moment by object's weight = $m \times 9.8 \times 0.3\,m$.
Anticlockwise moment by applied force = $15\,N \times 0.6\,m = 9\,Nm$.
Balancing both: $m \times 9.8 \times 0.3 = 9$ → $m \times 2.94 = 9$ → $m = 9/2.94 ≈ 3.06\,kg$.

Key Concept: Definition of Machine

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a bicycle can be considered as a machine. This is true as a bicycle meets the basic definition of a machine being a mechanical device that transmits and modifies human energy (pedaling) to produce motion. The reason correctly explains why a bicycle qualifies as a machine by providing the fundamental definition of what constitutes a machine - a device that transmits or modifies energy to perform work. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Basic Definition

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion (A) correctly defines a machine as a device that changes the direction or magnitude of a force to make work easier, which is true. However, the Reason (R) is false because not all machines necessarily increase energy efficiency; some may even lose energy due to friction. Thus, Assertion is true, but Reason is false.

Key Concept: Definition of a Machine

c) A system that applies mechanical power and has components performing specific tasks to achieve a purpose
[Solution Description]
A machine is a device that uses mechanical power and has multiple parts, each with a definite function, to perform a particular task. It typically converts input energy into useful output energy.

Key Concept: Function of Machine

b) Reduce workload by modifying force
[Solution Description]
Machines are designed to make work easier by altering the direction or magnitude of force applied. They help in reducing effort while increasing efficiency.

Key Concept: Principle of Moments, Equilibrium

b) 200 N
[Solution Description]
According to the principle of moments, for equilibrium:
$W_1 \times l_1 = W_2 \times l_2$
Here, $W_1 = 300 \, \text{N}$, $l_1 = 1 \, \text{m}$, $l_2 = 1.5 \, \text{m}$. Solving for $W_2$:
$300 \times 1 = W_2 \times 1.5 \implies W_2 = \frac{300}{1.5} = 200 \, \text{N}$

Key Concept: Couple in Rotation, Moment of Force

c) 200 N
[Solution Description]
Using the moment of force formula rearranged to solve for $F$:
$\tau = F \times r \implies F = \frac{\tau}{r}$
Here, $\tau = 60 \, \text{Nm}$ and $r = 0.3 \, \text{m}$. Thus:
$F = \frac{60}{0.3} = 200 \, \text{N}$

Key Concept: Moment of force

c) To maximize the perpendicular distance from the hinge, reducing the force required
[Solution Description]
The turning effect of a force depends on both the magnitude of the force and its perpendicular distance from the axis of rotation. By placing the handle at the farthest point from the hinges, the perpendicular distance is maximized. This allows a smaller force to produce the same moment of force required to open or close the door.
Moment of force = Force $\times$ Perpendicular distance.
Since distance is increased, less force is needed for the same effect.

Key Concept: Principle of Moments, Direction of Moment of Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a small force applied at the handle near the rim of a hand flour grinder produces enough torque to rotate the stone. This is true because torque ($\text{Torque} = F \times r$) increases with the perpendicular distance ($r$) from the axis of rotation. The reason explains that the moment of force is maximized when the perpendicular distance is largest, which directly justifies why applying force at the rim (largest distance) makes it easier to rotate the stone. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Moment of Force Calculations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Solution:
1) The assertion states that if two equal forces act upwards at both ends of a uniform rod pivoted at its midpoint, the resultant moment is zero. Let's verify this:
Moment due to force at end $A$ = $F \times 2\,\text{m}$ (clockwise).
Moment due to force at end $B$ = $F \times 2\,\text{m}$ (anticlockwise).
Net moment = $(F \times 2) - (F \times 2) = 0$. Thus, the assertion is true.
2) The reason states that for equilibrium, the net moment must be zero, which is correct as per the principle of moments.
3) Moreover, the reason correctly explains why the net moment is zero in this scenario. Hence, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Moment of Force Calculation

c) 40 N m
[Solution Description]
Moment of force = Force × Perpendicular distance.
Given: $F = 8\, \text{N}$, $d = 5\, \text{m}$.
Calculation: $8 \times 5 = 40\, \text{N m}$ (anticlockwise).

Key Concept: Principle of Moments, Equilibrium

b) $20\, \text{cm}$ right of $O$
[Solution Description]
For equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments. Let $x$ be the distance from $O$ for the $80\, \text{gf}$ weight.
- Anticlockwise moment due to $40\, \text{gf}$: $40 \times 40 = 1600\, \text{gf cm}$.
- Clockwise moment due to $80\, \text{gf}$: $80 \times x$.
Setting them equal: $80x = 1600 \implies x = 20\, \text{cm}$.
So, the $80\, \text{gf}$ weight must be placed $20\, \text{cm}$ to the right of $O$.

Key Concept: Resultant Moment of Forces

c) 45 N m
[Solution Description]
Each force creates a moment: $F \times d$. Here, $d = 1.5\, \text{m}$ (half of 3 m).
Upward force moment: $15 \times 1.5 = 22.5\, \text{N m}$ (clockwise).
Downward force moment: $15 \times 1.5 = 22.5\, \text{N m}$ (clockwise).
Resultant moment: $22.5 + 22.5 = 45\, \text{N m}$ (clockwise).

Key Concept: Moment of Force and Couple

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a longer-handled spanner requires less force to loosen or tighten a nut, which is true because the moment of force ($\tau$) is given by $\tau = F \times d$, where $F$ is the force and $d$ is the perpendicular distance from the axis of rotation. Increasing $d$ (by using a longer handle) reduces the required $F$ for the same $\tau$. The reason correctly explains that the moment of force depends on both the force and the perpendicular distance, which directly supports the assertion. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Moment of Force, Applications in Daily Life

b) 16 Nm
[Solution Description]
The moment of force is given by the product of the force and the perpendicular distance from the pivot. Here, the force is 20 N and the distance is 0.8 m. So, the moment of force is:
$M = F \times d = 20 \text{ N} \times 0.8 \text{ m} = 16 \text{ Nm}$
This moment acts anticlockwise if the force is applied outward.

Key Concept: Moment of Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that a longer spanner makes it easier to loosen a tight nut, which is true because a longer handle increases the perpendicular distance ($r$) from the pivot (nut). According to the formula for moment of force $\tau = F \times r$, a larger $r$ means less force ($F$) is required to achieve the same torque ($\tau$). The Reason correctly explains this by stating that the moment of force increases with the perpendicular distance from the pivot point. Thus, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.

Key Concept: Couple and its application

b) It decreases the amount of force needed
[Solution Description]
A couple consists of two equal and opposite parallel forces that produce rotation. For a spanner, the torque $\tau$ is given by $\tau = F \times d$, where $F$ is the force applied and $d$ is the perpendicular distance (handle length). Increasing $d$ increases the torque for the same force, making the spanner more effective.

Key Concept: Steering Wheel

b) Decrease the moment of force needed to turn the wheel
[Solution Description]
A larger steering wheel increases the perpendicular distance from the axis of rotation, enabling the driver to turn the wheel with minimal effort using the formula $Moment = F \times r$.

Key Concept: Examples of tools and devices using moments

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a longer handle on a spanner reduces the force needed to tighten a nut. This is true because the moment of force ($\tau = F \times r$) depends on both the applied force ($F$) and the perpendicular distance ($r$). Increasing $r$ (by using a longer handle) allows a smaller $F$ to produce the same moment.
The reason states that the moment of force increases with the perpendicular distance from the pivot point. This is also true, as evident from the formula $\tau = F \times r$.
Moreover, the reason correctly explains the assertion because it mathematically justifies why a longer handle (greater $r$) reduces the required force ($F$) for the same moment.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Moments and torque, spanner application

c) 600 N
[Solution Description]
To find the required force, we use the torque formula $\tau = F \times r$.
Given:
$\tau$ = 150 N$\cdot$m
$r$ = 25 cm = 0.25 m
Rearranging for $F$:
$F = \tau / r = 150 / 0.25 = 600$ N
Therefore, the minimum force needed is 600 N.

Key Concept: Door Handle

d) To maximize the moment of force for easier operation
[Solution Description]
Placing the handle far from the hinges maximizes the perpendicular distance ($d$) from the pivot point, allowing a small force ($F$) to produce a sufficient moment ($Moment = F \times d$) to open or close the door.

Key Concept: Moment of force, Principle of Moments

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this problem, we apply the Principle of Moments, which states that for a system in equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments.
Step 1: Identify the given values.
- Clockwise moment (right side): Force = 50 N, Distance = 0.5 m
- Anticlockwise moment (left side): Force = W, Distance = 1 m
Step 2: Calculate the clockwise moment:
$\text{Clockwise moment} = 50 \, \text{N} \times 0.5 \, \text{m} = 25 \, \text{N}\cdot\text{m}$
Step 3: Equate the clockwise and anticlockwise moments to find W:
$25 \, \text{N}\cdot\text{m} = W \times 1 \, \text{m}$
Step 4: Solve for W:
$W = 25 \, \text{N}$
Conclusion: The system is in equilibrium as the condition $\text{Clockwise moment} = \text{Anticlockwise moment}$ is satisfied. The Principle of Moments correctly explains why the system remains balanced when the products of forces and their respective distances are equal on both sides of the pivot.

Key Concept: Direction of Torque

b) Negative
[Solution Description]
Conventionally, if the effect on the body is to turn it clockwise, the moment of force is called the clockwise moment and is taken as negative.

Key Concept: Measurement of Moment of Force

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that applying a tangential force to a wheel maximizes the moment of force because the perpendicular distance equals the radius. This is true because for a wheel, the perpendicular distance from the pivot (center) to the tangent is indeed the radius. The Reason states that the moment of force depends on both the magnitude of the force and the perpendicular distance from the pivot, which is also true. Furthermore, the Reason correctly explains why the Assertion is valid, as it highlights the dependence of moment on the perpendicular distance, which is maximized in this case.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Torque, Vector Quantity

b) 16 Nm
[Solution Description]
Torque is given by the cross product of $\vec{r}$ and $\vec{F}$. Compute $\vec{\tau} = \vec{r} \times \vec{F} = (3\hat{i} + 4\hat{j}) \times (5\hat{i} + 12\hat{j})$. The cross product simplifies to $(3 \times 12) - (4 \times 5) = 36 - 20 = 16$ Nm in the $\hat{k}$ direction.

Key Concept: Principle of Moments

c) The sum of clockwise moments must equal the sum of anticlockwise moments
[Solution Description]
For a body to be in rotational equilibrium, the algebraic sum of moments about any point must be zero, meaning the clockwise moments balance the anticlockwise moments.

Key Concept: Conditions for Equilibrium

c) The sum of clockwise moments must equal the sum of anticlockwise moments.
[Solution Description]
Rotational equilibrium is achieved when the algebraic sum of moments of all forces acting on the body about the point of rotation is zero. This implies that the clockwise moments must balance the anticlockwise moments.

Key Concept: Static Equilibrium

d) The weight of the book and the normal force are equal in magnitude and opposite in direction.
[Solution Description]
The book is in static equilibrium because the weight of the book (downward force) is balanced by the normal force (upward force) exerted by the table. There is no net force acting on the book, so it remains at rest.

Key Concept: Static and Dynamic Equilibrium

d) Assertion is false, but Reason is true.
[Solution Description]
The book lying on the table is not moving; hence, it is in static equilibrium. Therefore, the Assertion (A) is false. However, the Reason (R) is true because for a body to be in dynamic equilibrium, the net force acting on it must indeed be zero. Thus, Assertion is false but Reason is true.

Key Concept: Translational Equilibrium

c) 50 N
[Solution Description]
The box is in translational equilibrium, so the net force acting on it is zero. The forces acting are:
Weight of the box ($W$) downward: $W = m \times g = 5 \times 10 = 50 \, \text{N}$
Normal force ($N$) upward balances the weight:
$N = W = 50 \, \text{N}$

Key Concept: Center of Gravity, Static Equilibrium

c) 45°
[Solution Description]
Step 1: Find center of gravity coordinates (x̄,ȳ):
Step 2: Area A1=1m² at (1,0.25), A2=0.75m² at (0.25,1.25)
Step 3: $x̄ = (1×1+0.75×0.25)/1.75 = 0.68m$
Step 4: $ȳ = (1×0.25+0.75×1.25)/1.75 = 0.68m$
Step 5: Toppling occurs when CG projects outside base support (2m wide)
Step 6: Critical angle θ when $x̄tanθ = ȳ$ → $0.68tanθ = 0.68$ → $θ = 45°$

Key Concept: Equilibrium Conditions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The book lying on the table remains at rest because the downward gravitational force (weight) is balanced by the upward normal force from the table, resulting in zero net force (translational equilibrium). Additionally, there is no rotational motion, so the net moment about any point is also zero (rotational equilibrium). Thus, both conditions for static equilibrium are satisfied, and the Reason correctly explains the Assertion.

Key Concept: Equilibrium Conditions, Principle of Moments

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that a uniform ladder leaning against a frictionless wall will remain in equilibrium if the net torque about its base is zero. This is true because equilibrium requires no rotational motion, which is ensured by zero net torque. However, the Reason states that for a body to be in equilibrium, both net force and net torque must be zero, which is a general condition for equilibrium. While the Reason is correct, it does not directly explain why the specific case of the ladder (Assertion) holds true, as the Assertion focuses only on the torque condition without mentioning the force condition. Thus, both are true, but the Reason is not the correct explanation of the Assertion.

Key Concept: Centripetal Force

b) Electrostatic force
[Solution Description]
The centripetal force required for an electron to move in a circular path around the nucleus is provided by the electrostatic force of attraction between the negatively charged electron and the positively charged nucleus.

Key Concept: Uniform Circular Motion

b) Because the direction of velocity continuously changes
[Solution Description]
Uniform circular motion is considered accelerated motion because although the speed remains constant, the direction of the velocity continuously changes. This change in direction implies a continuous change in velocity, which constitutes acceleration.

Key Concept: Centripetal and Centrifugal Force

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that centrifugal force is a real force, which is false because according to the syllabus, centrifugal force is a fictitious force, not a real one. The Reason states that centrifugal force is experienced by an observer moving in a circular path along with the body, which is true as it explains why centrifugal force is considered (to account for the apparent motion observed from a rotating frame of reference). Therefore, Assertion is false, but Reason is true.

Key Concept: Centripetal force, Circular motion, Acceleration

c) It becomes four times
[Solution Description]
The centripetal acceleration $a_c$ is given by $a_c = \frac{v^2}{r}$. If the speed $v$ is doubled, the new acceleration becomes $a_c' = \frac{(2v)^2}{r} = \frac{4v^2}{r} = 4a_c$. Thus, the centripetal acceleration becomes four times the original value.