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Assertion: A couple does not exert a net force on an object even though it exerts a torque.
Reason: A couple is a pair of two forces with an equal magnitude but opposite directions acting simultaneously on a body in different lines of action.

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Important Questions on Rotational Mechanics

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Assertion: To unscrew a rusted nut, we need a pipe wrench with longer arm.

Reason :Wrench with longer arm reduces the required force applied on the arm

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Assertion: The condition of equilibrium for a rigid body is -
Translational equilibrium: F=0, (i.e. the sum of all external forces equal to zero.)
Rotational equilibrium :τ=0 (i.e. the sum of all external torques equal to zero.)
Reason: A rigid body must be in equilibrium under the action of two equal and opposite forces.
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Assertion: For the purpose of calculation of moment of inertia, the body’s mass can be assumed to be concentrated at its centre of mass.
Reason: The moment of inertia of the rigid body about an axis passing through its centre of mass is zero.
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Assertion: A sharp needle when balanced on an edge falls even under a small disturbance.
Reason: Small disturbance causes torque in the same direction as that of angular displacement.
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Assertion: A rotating body can be in stable and unstable equilibrium.
Reason: Moment of inertia is always different for the different axis of rotation.
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Assertion: Practically a body rolling on a surface will lose its complete KE
Reason: Friction will convert its KE into PE of atoms.
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Assertion: Torque on a body can be zero even if there is a net force on it. 
Reason: Torque and force on a body are always perpendicular.
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Assertion: Torque on a body may be non-zero even if the net force acting on it is zero.
Reason: Force does not depend on point of application, but torque depends on point of application.