Problem 327 - JEE Physics

Question

The velocity of a body of mass \(20 \mathrm{~kg}\) decrease from \(20 \mathrm{~ms}^{-1}\) to \(5 \mathrm{~ms}^{-1}\) in a distance of \(100 \mathrm{~m}\). Force on the body is (A) \(-27.5 \mathrm{~N}\) (B) \(-47.5 \mathrm{~N}\) (C) \(-37.5 \mathrm{~N}\) (D) \(-67.5 \mathrm{~N}\)

Step-by-Step Solution

Answer

The force exerted on the body is \(-37.5\,\text{N}\) (option C).

Step 1: Calculate the initial and final kinetic energy of the body

We can find the initial and final kinetic energy of the body using the formula \(KE = \dfrac{1}{2}mv^2\), where \(m\) is the mass and \(v\) is the velocity. The initial kinetic energy is: \(KE_i = \dfrac{1}{2}(20\,\text{kg})(20\,\text{ms}^{-1})^2 = 4000\,\text{J}\) The final kinetic energy is: \(KE_f = \dfrac{1}{2}(20\,\text{kg})(5\,\text{ms}^{-1})^2 = 250\,\text{J}\)

Step 2: Determine the work done by force

The work done by force can be calculated as the change in kinetic energy of the body, which is given by: \(W = KE_f - KE_i\) Substituting the values, we get: \(W = 250\,\text{J} - 4000\,\text{J} = -3750\,\text{J}\)

Step 3: Use the work-energy theorem to calculate the force

According to the work-energy theorem, the work done by force is equal to the product of force and displacement. Thus, we can write the equation as: \(W = F \cdot d \cdot cos(\theta)\), where \(F\) is the force, \(d\) is the distance, and \(\theta\) is the angle between the force and displacement. In this case, since the body is decelerating, the force acting on the body is in the opposite direction of the displacement. So, \(\theta=180^\circ\), and \(cos(180^\circ)=-1\) Now, we can rewrite the equation as: \(-3750\,\text{J} = F \cdot (100\,\text{m}) \cdot (-1)\) \(F = \dfrac{-3750\,\text{J}}{100\,\text{m}} = -37.5\,\text{N}\) So, the force on the body is \(-37.5\,\text{N}\), which corresponds to option (C).

Key Concepts

Understanding Kinetic Energy

When we talk about kinetic energy, we're referring to the energy that an object possesses due to its motion. It's calculated using the formula:

• \( KE = \dfrac{1}{2}mv^2 \)

Here, \( m \) is the mass of the object and \( v \) is its velocity. This means that the kinetic energy of a body is directly proportional to the square of its velocity and its mass.

In our exercise, initially, the body has a velocity of \( 20 \,\mathrm{ms}^{-1} \) and a mass of \( 20 \,\mathrm{kg} \). Plugging these into our formula for kinetic energy, we find the initial kinetic energy as \( 4000 \,\mathrm{J} \).

As the body slows down to \( 5 \,\mathrm{ms}^{-1} \), its kinetic energy decreases to \( 250 \,\mathrm{J} \). This change in kinetic energy is key in understanding how much work is done on or by the system as forces act upon it.

Calculating the Force

The calculation of force on an object involves determining how much energy is transferred as the object moves through a distance. In this problem, we need to calculate the force causing the deceleration of the body using the work-energy theorem.

The work-energy theorem states that the work done by the net force acting on an object is equal to the change in kinetic energy of the object. Mathematically, it is expressed as:

• \( W = KE_f - KE_i \)

For our problem, we find:

• \( W = 250 \,\mathrm{J} - 4000 \,\mathrm{J} = -3750 \,\mathrm{J} \)

This negative value indicates that energy is being taken away from the body, namely through the force acting opposite to the motion, which is typical for deceleration.

We then use the equation:

• \( W = F \cdot d \cdot \cos(\theta) \)

With the force in the opposite direction of displacement, \( \theta = 180^\circ \), and hence \( \cos(180^\circ) = -1 \). Solving this gives:

• \( F = \dfrac{-3750 \,\mathrm{J}}{100 \,\mathrm{m}} = -37.5 \,\mathrm{N} \)

Understanding Deceleration

Deceleration occurs when an object slows down, or its velocity decreases over time. This change is caused by a net force acting in the opposite direction of the object's motion.

In our scenario, the body's velocity decreases from \( 20 \,\mathrm{ms}^{-1} \) to \( 5 \,\mathrm{ms}^{-1} \) as it moves a distance of \( 100 \,\mathrm{m} \). The work-energy principle helps us understand how much force is applied to achieve this change in speed. When forces cause a reduction in kinetic energy, the object decelerates.

In practical terms:

• The deceleration is a result of negative acceleration.
• It tells us that the object experiences a slowing effect due to the force.
• In calculations involving deceleration, it's essential to consider energy changes and work done by forces.

Understanding deceleration in this context allows us to see how energy transfers through a system, especially when slowing down. This can be vital in scenarios involving vehicle braking, amusement park rides, or any situation where controlling the speed of moving objects is necessary.