Assertion (A): A particle moving at constant speed and constant magnitude of radial acceleration must be undergoing uniform circular motion.
Reason (R): In uniform circular motion speed cannot change as there is no tangential acceleration.
Assertion (A): Constant speed and constant magnitude of radial acceleration ((v^2/r)) imply constant radius ((r)), which defines uniform circular motion. So (A) is True.
Reason (R): In uniform circular motion, acceleration is purely centripetal (radial), with no component tangential to the path. Thus, speed remains constant. So (R) is True.
Reason (R) correctly explains why constant speed and constant radial acceleration magnitude lead to uniform circular motion by implying constant radius and absence of tangential acceleration.
Assertion (A): If a body is in state of uniform circular motion then its velocity and acceleration both are varying.
Reason (R): If magnitude of velocity is \(v\) and radius of uniform circular motion is \(r\) then magnitude of acceleration is \(v^2/r\).
In uniform circular motion, velocity (vector) and acceleration (vector) are varying due to changing direction. So (A) is true. The magnitude of centripetal acceleration is \(a_c = v^2/r\). So (R) is true. However, (R) describes the magnitude, not the reason for vector variation. Thus, (R) is not the correct explanation of (A).
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.
With constant tangential acceleration \(a_t\) and increasing speed \(v\), centripetal acceleration \(a_c = v^2/R\) increases. The angle \(phi\) between resultant and tangential acceleration follows \(tan\phi = a_c/a_t\), so \(\phi\) increases. Thus (A) is true. (R) correctly defines these accelerations, providing the basis for (A).
Assertion (A): The equation of motion can be applied only if the acceleration is along the direction of velocity and is constant.
Reason (R): In circular motion, if velocity is constant then its motion is called uniform circular motion.
Equations of motion (kinematic equations) are for constant acceleration, irrespective of its direction relative to velocity. So (A) is false. In uniform circular motion, speed is constant, but velocity (a vector) continuously changes direction, hence it is not constant. So (R) is false. Both are false.
Assertion (A): In uniform circular motion, angular acceleration is zero.
Reason (R): In uniform circular motion, acceleration is constant.
In uniform circular motion, angular speed is constant, making angular acceleration \(alpha = 0\). So (A) is true. However, linear acceleration (centripetal acceleration) continuously changes direction, so it is not constant. Thus (R) is false.
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).
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): In non-uniform circular motion, velocity vector and acceleration vector are not perpendicular to each other.
Reason (R): In non-uniform circular motion, particle has normal as well as tangential acceleration.
In non-uniform circular motion, there is both tangential and centripetal acceleration. The tangential acceleration is parallel to velocity, so the resultant acceleration is not perpendicular to velocity. Reason (R) correctly identifies the components of acceleration, explaining why (A) is true.
Assertion (A): If a body is in state of uniform circular motion then its velocity and acceleration both are varying.
Reason (R): If magnitude of velocity is \(v\) and radius of uniform circular motion is \(r\) then magnitude of acceleration is \(v^2/r\).
In uniform circular motion, speed is constant, but velocity (direction) and acceleration (direction) vary, making (A) true. Reason (R) gives the correct magnitude of centripetal acceleration \(a = v^2/r\), so (R) is true. However, (R) describes the magnitude, not why the vectors are varying, so it's not the correct explanation.
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.
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): The equation of motion can be applied only if the acceleration is along the direction of velocity and is constant.
Reason (R): In circular motion, if velocity is constant then its motion is called uniform circular motion.
Assertion (A) is false; kinematic equations apply for constant acceleration (vector), not necessarily along velocity. Reason (R) is false; if velocity (vector) is constant, it's rectilinear motion, not circular motion. In uniform circular motion, *speed* is constant, but velocity changes direction.