Assertion: A body is in translational equilibrium if the net force on it is zero.
Reason: A body is in rotational equilibrium if the net torque on it about any one point becomes zero.
Translational equilibrium requires net force to be zero. Rotational equilibrium requires net torque to be zero. Both statements are true but independent definitions.
Consider the following statements \(A\) and \(B\), and identify the correct answer:
**Statement A:** In a perfectly rigid body, the net positive work done by external torques increases the rotational kinetic energy of the body.
**Statement B:** Angular acceleration of a rotating body having fixed axis of rotation is inversely proportional to the moment of inertia of the body for a given torque.
Statement A is correct because the rotational work-energy theorem states that work done equals change in rotational kinetic energy. Statement B is correct since \(alpha = frac{tau}{I}\), meaning angular acceleration is inversely proportional to the moment of inertia.
The force acting on a particle is \( \vec{F} = \hat{i} + 2\hat{j} + 3\hat{k} \text{ N} \). Find the torque (in N m) of this force about origin if position vector of the particle is \( \vec{r} = 7\hat{i} + 3\hat{j} + 5\hat{k} \text{ m} \).
Torque \( \vec{\tau} \) is given by \( \vec{r} \times \vec{F} \). Evaluating the cross product: \( \vec{\tau} = \det \begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \ 7 & 3 & 5 \ 1 & 2 & 3 \end{pmatrix} = -\hat{i} - 16\hat{j} + 11\hat{k} \text{ N m} \).
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 cyclist always bends inwards while negotiating a curve
Reason (R): By bending he lowers his centre of gravity
A cyclist bends inwards to provide the necessary centripetal force and maintain rotational equilibrium by balancing torques. Thus, (A) is true.
Lowering the center of gravity is not the primary reason for bending, making (R) false.
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.
Assertion (A): It is more difficult to open the door by applying the force near the hinge.
Reason (R): Torque is maximum at hinge.
Assertion (A) is true: Torque \(\tau = rF\sin\theta\) requires a larger force \(F\) for a smaller lever arm \(r\) (near the hinge). Reason (R) is false: Torque is zero at the hinge (pivot point) as \(r=0\). Thus, (A) is true, (R) is false.