Class 11 Physics: Work, Energy, and Power (Chapter 5) – Detailed Notes

11. Potential Energy

Potential energy is the energy a body possesses due to its position or configuration within a field. This energy changes with the configuration of the system. The two primary types of potential energy are gravitational and elastic.

11.1 Gravitational Potential Energy

Gravitational potential energy is the energy a body has due to its height above the earth’s surface. To calculate it, consider:

• m: mass of the body
• g: acceleration due to gravity
• h: height above the reference point

For small heights where g is nearly constant, the force required to counteract gravity is $F=mg$. The work done to lift the body to height h is:

$$\text{Work done}=F\times h=mgh$$

This work is stored as potential energy. The gravitational potential energy $V(h)$ is:

$$\text{Gravitational PE}=V(h)=mgh$$

11.2 Potential Energy of a Spring

Elastic potential energy is associated with the compression or extension of a spring. To compute it, consider an elastic spring fixed at one end with a mass m attached to the free end. Let:

• x: displacement from the spring’s equilibrium position
• k: spring constant

According to Hooke’s Law, the restoring force $F$ is:

$$F=-kx$$

The work done to stretch or compress the spring is:

$$dW=-Fdx=kxdx$$

To find the total work done, integrate from $x=0$ to $x=x$:

$$W={\int }_0^{x}kxdx=\frac{1}{2}k{x}^2$$

This work is stored as potential energy in the spring:

$$\text{Elastic PE}=\frac{1}{2}k{x}^2$$

12. Mechanical Energy and Its Conservation

Mechanical energy $E$ is the sum of kinetic energy $K$ and potential energy $V$:

$$E=K+V$$

In a system with conservative forces, mechanical energy is conserved. For a small displacement $x$ under a conservative force $F$, the work-energy theorem states:

$$\mathrm{\Delta }K=F(x)\mathrm{\Delta }x$$

With conservative forces, the potential energy $V(x)$ is defined such that:

$$-\mathrm{\Delta }V=F(x)\mathrm{\Delta }x\text{or}\mathrm{\Delta }V=-F(x)\mathrm{\Delta }x$$

Adding both expressions:

$$\mathrm{\Delta }K+\mathrm{\Delta }V=0\text{or}\mathrm{\Delta }(K+V)=0$$

Thus, the total mechanical energy $E$ remains constant:

$$E=K+V=\text{constant}$$

12.1 Illustration of the Law of Conservation of Mechanical Energy

Consider a body of mass m falling freely under gravity from height h:

• At Height A:

  $$\text{KE}=0,\text{PE}=mgh$$
  $$\text{Total Energy}=mgh$$
  

• At Height C (when the body reaches the ground):

  $$\text{KE}=\frac{1}{2}m{v}^2=mgh$$
  $$\text{PE}=0$$
  $$\text{Total Energy}=mgh$$
  

• At Height B (intermediate height):

  $$\text{KE}=\frac{1}{2}m{v}_1^2=mgx$$
  $$\text{PE}=mg(h-x)$$
  $$\text{Total Energy}=mgh$$
  

In all cases, total energy $E$ is $mgh$.

13. Different Forms of Energy

Energy can manifest in various forms, including:

• Heat Energy: Energy due to the random motion of molecules. For example, work done by friction converts kinetic energy into heat.
• Internal Energy: Total energy of molecules, including their potential and kinetic energies.
• Electrical Energy: Energy due to the flow of electric current.
• Chemical Energy: Energy stored in chemical bonds, released during chemical reactions.
• Nuclear Energy: Energy from atomic nuclei, obtainable through fission or fusion.

14. Mass-Energy Equivalence

Einstein’s mass-energy equivalence principle states:

$$E=m{c}^2$$

where:

• E is the energy,
• m is the mass,
• c is the speed of light in vacuum.

Mass and energy are conserved as a unified entity called mass-energy.

15. Principle of Conservation of Energy

The total energy of an isolated system remains constant. This principle is fundamental in physics, though it cannot be formally proved, no violations have been observed.

16. Work Done by a Variable Force

For a force that varies with position, the work done from point A to B is:

$$W_{A\to B}={\int }_{X_A}^{X_B}F(x)dx$$

This is represented by the area under the force versus position graph.

17. Conservative and Non-Conservative Forces

17.1 Conservative Forces

A force is conservative if the net work done moving a mass between two points is independent of the path taken.

17.2 Non-Conservative Forces

Forces that do not meet the criteria for conservativeness, such as friction, are non-conservative.

17.3 Conservative Forces and Potential Energy

For a conservative force $F(x)$, there exists a potential energy function $U(x)$. The work done by the force is related to the change in potential energy:

$$F(x)\mathrm{\Delta }x=-\mathrm{\Delta }U$$
$$F(x)=-\frac{dU}{dx}$$

Integrating:

$$U(b)-U(a)=-{\int }_a^{b}F(x)dx$$

18. Dynamics of Circular Motion

18.1 Force on the Particle

In uniform circular motion, the centripetal force $F_c$ is:

$$F_c=\frac{m{v}^2}{r}$$

This force acts towards the center of the circle.

18.2 Analyzing Forces

Resolve forces into radial and tangential components:

• Radial force: $F_r=\frac{m{v}^2}{r}$
• Tangential force: $F_t=m{a}_t$

18.3 Non-Uniform Circular Motion

For non-uniform circular motion, such as a vertical circle, analyze forces in radial and tangential directions.

18.4 Conical Pendulum

A conical pendulum consists of a mass rotating in a horizontal circle attached to a fixed point via a string inclined with the vertical.

Equations for the system are:

$$T\cos \theta =mg$$
$$T\sin \theta =m\frac{v^2}{r}$$

Time period $T$ is:

$$T=2\pi \sqrt{\frac{l\cos \theta }{g}}$$

18.5 Motion in a Vertical Circle

For a mass tied to a string and moving in a vertical circle:

• At the Top: Minimum velocity for circular motion:

$$v_t=\sqrt{gl}$$

• At the Bottom: Minimum velocity at the bottom:

$$v_b=\sqrt{v_t^2+4gl}$$

The motion in a vertical circle involves varying speed and is an example of non-uniform circular motion