Assertion (A): The condition of equilibrium for a rigid body is – Translational equilibrium: \( \sum \vec{F} = 0 \) (i.e. sum of all external forces equal to zero). Rotational equilibrium: \( \sum \vec{\tau} = 0 \) (i.e. sum of all external torques equal to zero.)
Reason (R): A rigid body must be in equilibrium under the action of two equal and opposite forces.
For rigid body equilibrium, both net force and net torque must be zero. Assertion (A) correctly states this.
Reason (R) is false; two equal and opposite forces can form a couple if not collinear, causing rotation and thus not guaranteeing equilibrium.
Assertion (A): A wheel slides downward on frictionless inclined plane, without rolling.
Reason (R): In pure rolling work done against friction always zero.
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.
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).
Assertion (A): A solid copper and solid aluminium sphere of same masses are spinning about their axes with same angular velocities copper sphere has more angular momentum than aluminium.
Reason (R): Both copper and aluminium sphere have same radius.
Angular momentum \( L = I\omega = \frac{2}{5} MR^2 \omega \). If mass (M) and angular velocity (\( \omega \)) are the same, L depends on \( R^2 \). Copper is denser than aluminium (\( \rho_{Cu} > \rho_{Al} \)). For the same mass, \( V_{Cu} < V_{Al} \), implying \( R_{Cu} < R_{Al} \). Therefore, \( L_{Cu} < L_{Al} \), making (A) false.
Also, \( R_{Cu} < R_{Al} \), so (R) is false. Both (A) and (R) are false.
Assertion (A): When a sphere is rolls on a horizontal table it slows down and eventually stops.
Reason (R): When the sphere rolls on the table, both the sphere and the surface deform near the contact. As a result, the normal force does not pass through the centre and provide an angular deceleration.
A rolling sphere stops due to rolling friction. This friction arises from deformations at the contact point, causing the normal force to produce a torque that opposes the rolling motion, leading to angular deceleration.
Assertion (A): A ladder is more likely to slip when a person is near the top than when he is near the bottom.
Reason (R): The friction between the ladder and floor decreases as he climbs up.
Assertion (A) is true; a ladder is more likely to slip when a person is higher up due to increased outward horizontal thrust. Reason (R) is false; the normal force at the base remains constant, so the maximum static friction available does not decrease.