Question

STATEMENT-1 : Work done by a force on a body whose centre of mass does not move may be non-zero. STATEMENT-2 : Work done by a force depends on the displacement of the centre of mass. (1) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (2) Statement- 1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement- 1 is True, Statement-2 is False (4) Statement-1 is False, Statement-2 is True.

Short answer

Option (3)

Step-by-step solution

Understand Statement-1

Statement-1 says that the work done by a force on a body whose center of mass does not move may be non-zero. This implies that even if the center of mass of a body remains stationary, there can still be work done, such as in the case of rotational motion where the center of mass might not move but different parts of the object still move.

Understand Statement-2

Statement-2 says that the work done by a force depends on the displacement of the center of mass. According to the definition of work, work done by a force is given by \(W = F \times d \times \cos(θ)\), where \(d\) is the displacement of the point of application of the force (which can be considered as the center of mass for translational motion).

Evaluate the Truth of Each Statement

Statement-1 is true because work can indeed be done on a body even if its center of mass does not move, as seen in rotational motion. Statement-2 is false because work does not necessarily depend on the displacement of the center of mass but on the displacement of the point of application of the force.

Determine the Correct Answer

Now that we have established that Statement-1 is true and Statement-2 is false, the correct answer is option (3).

Key concepts

Work Done by Force

Let's delve into the core concept of 'Work Done by Force'. Work is defined as the product of the force applied on an object and the displacement of that object in the direction of the force. Mathematically, it can be expressed as: \[ W = F \times d \times \text{cos}(\theta) \] where: * \( W \) is the work done by the force * \( F \) is the magnitude of the force * \( d \) is the displacement * \( \theta \) is the angle between the force and the direction of displacement In the context of the given problem, it's very important to note that the work done depends on the displacement of the object or the point of application of the force. This means even if the center of mass doesn't move, like in some cases of rotational motion, different parts of the object may still move, resulting in non-zero work done.

Center of Mass

The center of mass is a crucial point in a body or system of bodies where the entire mass can be considered to be concentrated for certain calculations. In simple terms, it's the average location of all the mass in an object. One can find the center of mass using the formula: \[ \vec{R} = \frac{1}{M} \sum_{i} m_{i} \vec{r}_{i} \] where: * \( \vec{R} \) is the position vector of the center of mass * \( M \) is the total mass of the object * \( m_{i} \) is the mass of individual particles * \( \vec{r}_{i} \) is the position vector of individual particles For an object under rotational motion, the center of mass might remain stationary while the object rotates around it. Importantly, forces applied off-center can cause objects to rotate, providing a clear instance where work can be done (rotational motion), even if the center of mass doesn't move.

Rotational Motion

Rotational motion involves the movement of a body around a given axis. When a force is applied to a point on the object that is not at its center or along the axis of rotation, it causes the object to rotate. The work done in rotational motion can be described in terms of torque and angular displacement: \[ W = \tau \times \theta \] where: * \( \tau \) is the torque * \( \theta \) is the angular displacement Torque is the rotational equivalent of force and can be expressed as: \[ \tau = r \times F \times \text{sin}(\phi) \] where: * \( r \) is the lever arm length from the axis of rotation * \( F \) is the applied force * \( \phi \) is the angle between the force and the lever arm In applications involving rotational motion, even if the center of mass remains fixed, different parts of the body are displaced, and therefore, work is done. This intric exposure clears the confusion in the given problem where it's assumed that work can only be related to the movement of the center of mass.