Assertion (A): A body tied to an end of a string is whirled along a vertical circle with such a velocity at the lowest point that, at some position, tension in the string is zero but the speed at the position is non-zero. The body will leave the circular path at the position of zero tension.
Reason (R): In vertical circular motion, so as to cross the highest point along the circle, speed at the highest point, \( v_H geq 0 \).
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
An object leaves a vertical circular path when tension becomes zero and speed is non-zero. So (A) is true. To complete a vertical circle, the minimum speed at the highest point is \( \sqrt{gR} \), not just \( 0 \). So (R) is false.
Assertion (A): When an automobile while going too fast around a curve overturns, its inner wheels leave the ground first.
Reason (R): The inner wheels are moving in a circle of smaller radius, the maximum permissible velocity for them is less.
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: Overturning occurs when the centrifugal force moment exceeds the stabilizing moment, causing the inner wheels to lift.
Reason (R) is true: The maximum safe velocity \( v_{\text{max}} = \sqrt{mu gr} \), so a smaller radius \( r \) means a smaller \( v_{\text{max}} \). However, (R) does not explain the overturning mechanism itself. Thus, (R) is not the correct explanation of (A).
Assertion (A): On an unbanked road, as the frictional force increases, the safe velocity limit for taking a turn also increases.
Reason (R): Banking of roads will increase the value of limiting velocity.
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: On an unbanked road, the maximum safe velocity is \( v_{\text{max}} = \sqrt{mu_s gr} \). An increase in friction (\( \mu_s \)) leads to an increased \( v_{\text{max}} \).
Reason (R) is true: Banking of roads provides a component of the normal force for centripetal force, effectively increasing the limiting velocity. (R) is not the correct explanation for (A) as they represent different factors influencing safe velocity.
Assertion (A): A coin is placed on the gramophone. When the motor starts, the coin moves along the gramophone. As the speed goes on increasing, the coin flies off after some time.
Reason (R): The gravitational force of gramophone provides the necessary centripetal force to the coin.
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 the angular speed \( \omega \) of the gramophone increases, the required centripetal force \( m \omega^2 r \) for the coin increases. When this force exceeds the maximum static friction (\( \mu_s mg \)), the coin slips off.
Reason (R) is false: The centripetal force is provided by the frictional force between the coin and the gramophone, not by the gravitational force. Gravitational force provides the normal force.
Assertion (A): Two identical trains move in opposite sense in equatorial plane with equal speed relative to earth’s surface. They have equal magnitude of normal reaction.
Reason (R): The trains require same centripetal force although they have different speeds.
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
The absolute speed of the trains relative to Earth's center is \( v_{\text{abs}} = R_E \omega_E pm v_{\text{surface}} \). Since their absolute speeds are different, the centripetal force required \( F_c = m v_{\text{abs}}^2 / R_E \) will be different. The normal reaction is \( N = mg - F_c \), so it will also be different. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): The work done by the net force on a particle during non-uniform circular motion is not equal to zero.
Reason (R): In case of non-uniform circular motion net force and elementary displacement are not perpendicular to each other.
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
In non-uniform circular motion, there is a tangential component of force, which causes a change in speed. Work done is `\(W = \int \vec{F}_{net} \cdot d\vec{r}\)`.
Since the net force is not always perpendicular to the elementary displacement `\(d\vec{r}\)` due to the tangential component, the work done by the net force is not zero. Both assertion and reason are true, and the reason correctly explains the assertion.