Work Done by Variable Force: Comprehensive NEET Physics Notes

1. Work Done by Variable Force

1.1 Definition and Concept

In real-life scenarios, the forces encountered are rarely constant. Instead, they often vary depending on factors such as position, time, or the object’s motion. For example, pulling a spring, air resistance, and friction are all examples of variable forces.

When dealing with a variable force, the work done is not simply the product of force and displacement as with constant forces. Instead, we must sum the infinitesimal contributions to the work over each small segment of the motion. The total work done by a force that varies with position, $F(x)$, is given by the integral:

$W={\int }_{x_i}^{x_f}F(x),dx$

Here, $x_i$ and $x_f$ represent the initial and final positions. This integral represents the area under the curve of the force function over the displacement.

NEET Tip: When solving NEET problems related to variable forces, ensure you use proper limits of integration. The correct bounds of the integral represent the starting and ending points of the displacement, which is crucial for accurate calculation of work.

1.2 Graphical Interpretation

A variable force can be visualized using a graph of $F(x)$ versus $X$. The area under the curve between two points gives the work done over that displacement. If the force changes with position, the work done is represented by the total area between the curve and the x-axis.

• Positive Work: When the force acts in the same direction as the displacement, the area under the curve contributes to positive work.
• Negative Work: When the force opposes the displacement, the work done is negative, and the area under the curve lies below the x-axis.

Did You Know?
Variable forces are often encountered in nature, such as the force of friction changing with speed or the force exerted by a spring that stretches or compresses. These real-life examples show why integration is necessary to determine the total work done accurately.

1.3 Work Done by a Non-Uniform Force: Example

Let’s consider an example where the force varies with position. Suppose a force is given by $F(x)=kx$, where $k$ is a constant. To calculate the work done by this force as the object moves from position $x_i$ to $x_f$, we set up the following integral:

$W={\int }_{x_i}^{x_f}kxdx$

This simplifies to:

$W=\frac{k}{2}(x_f^2-x_i^2)$

This result shows that the work done by the force depends on the change in the square of the displacement, making it distinct from the work done by constant forces.

Common Misconception:
It is a common misunderstanding that work done by a variable force is simply the product of force and displacement. For variable forces, it is essential to use the integral to calculate work correctly, as the force changes over the path.

1.4 Work Done by a Spring

A common example of a variable force is the restoring force exerted by a spring, which follows Hooke's Law. The force exerted by a spring is proportional to the displacement from its equilibrium position, and is given by:

$F(x)=-kx$

The work done by a spring as it is compressed or stretched from position $x_i$ to $x_f$ is calculated as:

$W={\int }_{x_i}^{x_f}-kxdx=\frac{k}{2}(x_i^2-x_f^2)$

In this case, the work done by the spring is negative, as the force exerted by the spring opposes the displacement.

Real-Life Application:
Springs are widely used in mechanical systems such as car suspensions and watches. The variable force applied by a spring allows precise control of motion, making it a key component in various devices.

Quick Recap

• Work done by a variable force is given by the integral:
  $W=\int F(x)dx$
• Graphically, the work is represented by the area under the force-displacement curve.
• For a linearly varying force, $F(x)=kx$, the work is given by:
  $W=\frac{k}{2}(x_f^2-x_i^2)$.
• For a spring, the work is:
  $W=\frac{k}{2}(x_i^2-x_f^2)$.

NEET Problem-Solving Strategy:

• Break down variable force problems into small steps or intervals and solve using integration.
• Pay attention to the direction of the force and displacement; negative work occurs when the force opposes displacement.
• Use graphical interpretation when possible to visualize work as the area under the force curve, which aids in understanding complex problems.

Concept Connection:

This concept connects with potential energy in physics. In both cases, the work done by a variable force can be stored as potential energy, which is released as kinetic energy. Understanding variable forces also has connections in chemistry, where potential energy plays a significant role in understanding chemical bonds and reactions.

Practice Questions:

1. A force $F(x)=2x^2$ acts on a particle moving along the x-axis from $x=0\text{m}$ to $x=3\text{m}$. Calculate the work done by this force.
2. • Solution:
     $W={\int }_0^{3}2x^{2}dx={\left[\frac{2x^3}{3}\right]}_0^3=54\text{J}$
2. A spring with a spring constant of $k=100\text{N/m}$ is compressed from $x=0.5\text{m}$ to $x=0.0\text{m}$. Calculate the work done by the spring.
3. • Solution:
     $W=\frac{k}{2}(x_i^2-x_f^2)=\frac{100}{2}(0.5^2-0^2)=12.5\text{J}$
3. A varying force $F(x)=4x-2$ acts on a body. Calculate the work done by the force as the body moves from $x=0$ to $x=3\text{m}$.
4. • Solution:
     $W={\int }_0^3(4x-2)dx=2x^2-2x{|}_0^{}$