Problem 20 - Master Resource Book in JEE Main Physics

Question

A cord is used to lower vertically a block of mass \(M\) by a distance \(d\) with constant downward acceleration \(g / 4\) work done by the cord on the block is (a) \(M g \frac{d}{4}\) (b) \(3 M g \frac{d}{4}\) (c) \(-3 M g \frac{d}{4}\) (d) \(\mathrm{Mgd}\)

Step-by-Step Solution

Answer

(b) \(3Mg \frac{d}{4}\)

Step 1: Identify Known Quantities

The block of mass \(M\) is lowered by a distance \(d\) with a constant downward acceleration \(g/4\).

Step 2: Write the equation of motion

The net force on the block can be expressed using Newton's second law \( F = ma \). So,\[ M(g - a) = T - Mg \]where \(T\) is the tension in the cord. Given acceleration \(a = g/4\),\( M \left( g - \frac{g}{4} \right) = Mg - T \).

Step 3: Calculate the net force

The force exerted by the tension is:\[ T = M \left( g - \frac{3g}{4} \right) = \frac{3Mg}{4} \]

Step 4: Determine Work Done by the Cord

The work done \(W_T\) by the tension in the cord as it moves the block down a distance \(d\) is given by:\[ W_T = T \cdot d \]\[ = \frac{3Mg}{4} \cdot d \]

Step 5: Conclusion

The work done by the cord on the block is \(3Mg \frac{d}{4}\). Hence, the correct choice is (b) \(3Mg \frac{d}{4}\).

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that describes how objects behave when subjected to forces. It states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. In mathematical form, it is expressed as:\[ F = ma \]Where:

• \(F\) is the net force acting on the object
• \(m\) is the mass of the object
• \(a\) is the acceleration of the object

This law reveals that an object's acceleration depends directly on the net force and inversely on its mass.
It means heavier objects require more force to achieve the same acceleration as lighter ones.
In the exercise, when the block is lowered, the forces acting are gravity and the tension in the cord.To find the net force, you should sum up these forces, taking into account their directions.
The tension force, however, acts in opposition to gravity, making it essential to write the correct equation of motion for the system.

Tension in a Cord

The tension in a cord is the force exerted along the cord when it is pulled tight by forces acting at either end.
This force opposes the gravitational force acting on the block in the exercise.To calculate the tension, we utilize the equation of motion derived from Newton's second law:\[ M(g - a) = T - Mg \]Here, \(M\) is the mass, \(g\) is gravitational acceleration, and \(a\) is the given acceleration.
The tension \(T\) can thus be solved by rearranging the equation:\[ T = M \left(g - \frac{3g}{4} \right) \]\[ T = \frac{3Mg}{4} \]Thus, the tension in the cord is essentially measuring the force needed to produce a downward acceleration distinct from free-fall due to gravity.
It is less than the total weight of the block due to the block's reduced acceleration.

Acceleration

Acceleration is defined as the rate of change of velocity of an object with respect to time. In scenarios involving gravity, like in the exercise, gravity generally provides a constant acceleration to a freely falling object, denoted by \( g \), approximately \(9.8 \ ms^{-2} \).In the given exercise, the block experiences a downward acceleration of \( \frac{g}{4} \).
This means that its downward velocity is increasing at a quarter of the full gravitational acceleration.
This reduced acceleration is achieved through the cord's tension force acting less than the full weight of the block.The concept of acceleration further ties into energy and work calculations.
Knowing the acceleration helps determine the net force on the object, which allows for the calculation of work done by varying forces.