Given below are two statements:
Statement I: Centre of mass of any object always coincide with centre of gravity.
Statement II: Centre of gravity is the point where total gravitational torque on the body is zero.
In the light of the above statements, choose the most appropriate answer from the options given below.
1. Both statements I and II are correct
2. Both statements I and II are incorrect
3. Statement I is correct but II is incorrect
4. Statement I is incorrect but II is correct
View Answer
Statement I is false because the center of mass and center of gravity only coincide in a uniform gravitational field. Statement II is true because the center of gravity is defined as the point about which the net gravitational torque is zero.
Assertion (A): A body with negative energy cannot have linear momentum.
Reason (R): Magnitude of linear momentum can be negative.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Total mechanical energy \(E = K + U\) can be negative if potential energy \(U\) is negative and larger in magnitude than kinetic energy \(K\).
However, kinetic energy \(K = \frac{1}{2}mv^2\) is always non-negative, implying momentum exists. The magnitude of linear momentum \(|\vec{p}| = mv\) is always non-negative.
Therefore, both Assertion and Reason are false.
Assertion (A): In any kind of collision, kinetic energy cannot be same throughout.
Reason (R): In elastic collision kinetic energy remains constant throughout.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true as kinetic energy is not conserved in all types of collisions (e.g., inelastic collisions). Reason (R) is true as kinetic energy is conserved in elastic collisions. However, (R) does not correctly explain (A).
Assertion (A): In a perfectly inelastic collision there is a limit to the loss of kinetic energy of colliding bodies.
Reason (R): In perfectly inelastic collision, linear momentum of system is conserved.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Both (A) and (R) are true. In a perfectly inelastic collision, momentum is conserved (R), which allows the calculation of the final common velocity (\(v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}\)) and thus the minimum kinetic energy (\(KE_f = \frac{1}{2}(m_1+m_2)v_f^2\)) that must remain, placing a limit on kinetic energy loss (A). Hence, (R) correctly explains (A).
Assertion (A): Centre of mass of a body in pure rolling on a horizontal surface always moves in a straight line.
Reason (R): Centre of mass of a body must be inside the body.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true; the center of mass of a body in pure rolling on a horizontal surface moves along a straight line (rectilinear motion). Reason (R) is false; the center of mass can be outside the physical boundaries of the body (e.g., for a ring or a hollow sphere).
Assertion (A): In two particle system when viewed from center of mass reference frame, if one particle stops then other one will also stop simultaneously, irrespective of external forces acting on system.
Reason (R): Centre of mass of a system is a point about which total momentum of system is always constant and non-zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true.
In the center of mass (CM) frame, the total momentum of the system is always zero. If \(p_1 + p_2 = 0\), then \(p_1 = -p_2\). If \(p_1 = 0\), then \(p_2\) must also be zero.
Reason (R) is false; total momentum of the system about the center of mass is always zero, not 'constant and non-zero'.
Assertion (A): Two particles undergo rectilinear motion along different straight lines. Then the centre of mass of system of given two particles also always moves along a straight line.
Reason (R): If direction of net momentum of a system of particles (having non-zero net momentum) is fixed, the centre of mass of given system moves along a straight line.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Both (A) and (R) are true. The velocity of the center of mass is given by \(V_{CM} = \frac{P_{total}}{M_{total}}\). If particles move rectilinearly, their velocities are constant, making \(P_{total}\) constant in direction. A fixed direction of \(V_{CM}\) means rectilinear motion. Hence, (R) correctly explains (A).
Assertion (A): A half filled bottle is more stable than a fully filled identical bottle when kept in upright position.
Reason (R): A half filled bottle has lesser mass than a fully filled bottle. (The fluid and bottles are identical).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. (A) is true and (R) is false
View Answer
Assertion (A) is false because a fully-filled bottle has a lower center of gravity and lacks the destabilizing sloshing effect of a partially-filled bottle.
Reason (R) is true as a half-filled bottle naturally contains less mass
Assertion (A): Two objects are moving towards each other due to mutual attraction. The kinetic energy of the system remains constant.
Reason (R): Total linear momentum of the system consisting both the objects remain constant even in the presence of external forces.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is false; as objects move towards each other due to mutual attraction, their potential energy decreases, and kinetic energy increases, so it does not remain constant.
Reason (R) is false; total linear momentum of a system is conserved only in the absence of external forces, not 'even in the presence of external forces'. Therefore, both (A) and (R) are false.
Consider a one-dimensional head on collision of two balls.
Assertion (A): The loss in kinetic energy of the system during the collision does not depend on the velocity of the observer.
Reason (R): Kinetic energy of a body is independent of velocity of observer.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Assertion (A): The loss in kinetic energy of a system is generally dependent on the observer's frame of reference. Thus, (A) is false.
Reason (R): Kinetic energy \(K = \frac{1}{2}mv^2\) depends on the velocity (v\), which is relative to the observer. Therefore, kinetic energy is dependent on the velocity of the observer. Thus, (R) is false.
Since both (A) and (R) are false, option (4) is correct.