The figure shows a horizontal block of mass M suspended by two wires A and B. The centre of mass of the block is closer to B than A. (i) Is the magnitude of the torque due to wire A is greater, less or equal to that due to B w.r.t. centre of mass ? (ii) Which wire A or B exerts more force on the block ?
1. (i) greater (ii) B
2. (i) equal (ii) B
3. (i) less (ii) A
4. (i) greater (ii) A
View Answer
As the object is in rotational equilibrium, Net torque acting on the object is zero.
so, Torque of TA = Torque of TB
TAXA= TBXB
\[ \frac{T_{A}}{T_{B}}=\frac{X_{B}}{X_{A}} \]
\[ X_{B} < X_{A} \]
\[ T_{A} < T_{B} \]
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.
1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
2. Both Assertion and Reason are true but Reason is not correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. Assertion is false and Reason is true.
View Answer
Translational equilibrium requires net force to be zero. Rotational equilibrium requires net torque to be zero. Both statements are true but independent definitions.
A ladder of length \(\ell\) and mass m is placed against a smooth vertical wall but the ground is not smooth. Coefficient of friction between the ground and the ladder is \(\mu\). The minimum angle \(\theta\) with ground at which the ladder will stay in equilibrium is :
1. \(tan^{-1}(\mu)\)
2. \(tan^{-1}(2\mu)\)
3. \(tan^{-1}(\mu/2)\)
4. \(tan^{-1}(1/2\mu)\)
View Answer
For translational and rotational equilibrium of the ladder, taking torque about the base gives \(N_{\text{wall}} \ell \sin\theta = mg \frac{\ell}{2} \cos\theta\). With \(N_{\text{wall}} = f \le \mu mg\), we get the minimum angle for equilibrium to be \(\tan\theta = \frac{1}{2\mu}\).
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.
1. A is correct but B is incorrect
2. A is incorrect but B is correct
3. Both A and B are correct
4. Both A and B are incorrect
View Answer
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 \( \vec{r} = 7\hat{i} + 3\hat{j} + 5\hat{k} \text{ m} \).
1. \( \hat{i} + 16\hat{j} - 11\hat{k} \)
2. \( -\hat{i} - 16\hat{j} + 11\hat{k} \)
3. \( \hat{i} + 16\hat{j} + 11\hat{k} \)
4. \( -\hat{i} + 9\hat{j} + 11\hat{k} \)
View Answer
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.
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
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
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
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.
