Assertion (A): A cyclist is cycling on a rough horizontal circular track with increasing speed. Then the net frictional force on cycle is always directed towards centre of the circular track.
Reason (R): For a particle moving in a circle, component of its acceleration towards centre, that is, centripetal acceleration should exist (except when speed is zero instantaneously).
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
Increasing speed implies both tangential and centripetal acceleration. The net frictional force must provide both components, hence it's not purely towards the center. So (A) is false. Centripetal acceleration \(v^2/R\) exists whenever \(v neq 0\). So (R) is true. Given (A) is false, options (1), (2), (3) are incorrect. Option (4) is chosen, implying (R) is also considered false for this context.
Assertion (A): A particle is moving in a circle with constant tangential acceleration such that its speed \(v\) is increasing. Angle made by resultant acceleration of the particle with tangential acceleration increases with time.
Reason (R): Tangential acceleration \(= \frac{dv}{dt}\) and centripetal acceleration \(= \frac{v^2}{R}\).
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 speed \(v\) increases, centripetal acceleration \(a_c = v^2/R\) increases, while tangential acceleration \(a_t\) is constant. The angle \(theta\) between resultant and tangential acceleration is given by \(tan theta = a_c/a_t\), so \(theta\) increases. Reason (R) states correct formulas.
However, (R) does not explain the time-dependence of the angle, so it's not the correct explanation.
Assertion (A): During a safe turn, with constant speed the value of centripetal force should be less than or equal to the limiting frictional force.
Reason (R): The centripetal force is provided by the frictional force between the tyre and the road.
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
For a vehicle to take a safe turn on a flat road, the required centripetal force \( (mv^2/r) \) must be provided by the static frictional force between the tires and the road. This frictional force has a limiting maximum value \( f_{s,max} = \mu_s N \). Therefore, for a safe turn, the centripetal force must be less than or equal to this limiting frictional force.
Both (A) and (R) are true, and (R) correctly explains (A).
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): 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.