Assertion (A): Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The solid cylinder will reach the bottom of the inclined plane first.
Reason (R): By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.
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 acceleration of a rolling body down an incline is \( a = \frac{g\sin\theta}{1 + I/(MR^2)} \). For a solid cylinder, \( I = \frac{1}{2}MR^2 \); for a hollow cylinder, \( I = MR^2 \). Since the solid cylinder has a smaller \( I/(MR^2) \) ratio, its acceleration is greater, and it reaches the bottom first.
By conservation of energy, \( Mgh \) converts to kinetic energy, so total KEs are identical if \( M \) and \( h \) are same.
Thus A and R are true, but R does not explain why one reaches first (which depends on the distribution of KE).
Assertion (A): A wheel slides downward on frictionless inclined plane, without rolling.
Reason (R): In pure rolling work done against friction always 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
On a frictionless inclined plane, there's no torque to cause rotation, so a wheel will only slide; thus (A) is true.
In pure rolling, the point of contact is instantaneously at rest, so static friction does no work *by* it. The statement 'work done against friction always zero' is not universally true, making (R) false.
Assertion (A): If a sphere starts pure rolling down a rough incline plane, work done by friction is zero.
Reason (R): Work done by friction for translational motion is negative and work done by friction for rotational motion is positive and equal in magnitude.
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 pure rolling, the point of contact is instantaneously at rest, so static friction does no work (A is true).
Friction opposes translational motion (negative work) and causes rotation (positive work); these works are equal in magnitude, leading to zero net work (R is true and correctly explains A).