Q.Give example of a situation in which an applied force does not result in a change in kinetic energy.

The key idea is the Work-Energy Theorem, which states that the net work done on an object equals its change in kinetic energy.

1. The Work-Energy Theorem is given by $\Delta KE = W_{net}$.
2. For the kinetic energy ($\Delta KE$) not to change, the net work done ($W_{net}$) on the object must be zero.
3. Work done by a constant force $\vec{F}$ over a displacement $\vec{d}$ is $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$, where $\theta$ is the angle between $\vec{F}$ and $\vec{d}$. …

The change in an object's kinetic energy is directly related to the net work done on it. This fundamental relationship is described by the work-energy theorem.

Concept and Intuition

Kinetic energy is the energy an object possesses due to its motion, given by $KE = \frac{1}{2}mv^2$. A change in kinetic energy means a change in the object's speed. The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.

Here, $W_{net}$ is the total work done by all forces acting on the object, $KE_f$ is the final kinetic energy, and $KE_i$ is the initial kinetic energy.

For an applied force not to result in a change in kinetic energy, the net work done on the object must be zero ($\Delta KE = 0$). This can happen in a few ways:

1. The applied force does no work: Work done by a force is defined as $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$, where $\theta$ is the angle between the force vector ($\vec{F}$) and the displacement vector ($\vec{d}$). If $\theta = 90^\circ$ (force is perpendicular to displacement) or if $d = 0$ (no displacement), then the work done by that force is zero.
2. The work done by the applied force is cancelled by other forces: Even if an applied force does positive or negative work, if other forces do an equal amount of opposite work, the net work done on the object will be zero.

We are looking for a situation where an applied force is present, but it does not change the kinetic energy. The most direct way for this to happen is if the applied force itself does no work.

Example: Centripetal Force in Uniform Circular Motion

Consider an object moving in a circular path at a constant speed. This is known as uniform circular motion.

1. Identify the applied force: For an object to move in a circle, there must be a force directed towards the center of the circle. This force is called the centripetal force. For example, a string pulling a ball in a circle, or gravity keeping a satellite in orbit. This is our "applied force."

2. Analyze the direction of force and displacement:

   • The centripetal force ($\vec{F}_c$) is always directed radially inward, towards the center of the circle.
   • The instantaneous displacement ($\vec{d}$) of the object is always tangential to the circular path, in the direction of its velocity.

3. Determine the angle between force and displacement: At any instant, the centripetal force vector is perpendicular to the instantaneous displacement vector. Therefore, the angle $\theta$ between $\vec{F}_c$ and $\vec{d}$ is $90^\circ$.

4. Calculate the work done by the centripetal force:

   Using the formula for work done: …

Concept: Work–Energy Theorem

The work–energy theorem states $\Delta KE = W_{net}$. For kinetic energy to remain unchanged, the net work done on the body must be zero — this happens whenever the (only) applied force stays perpendicular to the body's velocity at every instant.

Step 1: State the condition needed

For $\Delta KE = 0$ with $F \neq 0$ and $d\neq0$, we need $\cos\theta = 0$, i.e. $\theta = 90^\circ$.

Step 2: Find a physical situation with the force always perpendicular to the velocity

In uniform circular motion, the centripetal force always points toward the centre, while the velocity (and the instantaneous displacement) is always tangential — perpendicular to the force at every instant. …