A-Level Mathematics 9709 - Mechanics: Energy, Work and Power

1. Introduction

This section explores the fundamental concepts of energy, work, and power in mechanics. We will cover kinetic and potential energy, the principle of conservation of energy, work done by forces, and the definition of power.

2. Kinetic and Potential Energy

2.1 Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion.

The formula for kinetic energy is:

$KE = \frac{1}{2}mv^2$

where:

• $KE$ is the kinetic energy (Joules, J)
• $m$ is the mass of the object (kg)
• $v$ is the velocity of the object (m/s)

2.2 Potential Energy

Potential energy is stored energy possessed by an object due to its position or configuration.

2.2.1 Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its height above a reference point.

The formula for gravitational potential energy is:

$PE = mgh$

where:

• $PE$ is the gravitational potential energy (Joules, J)
• $m$ is the mass of the object (kg)
• $g$ is the acceleration due to gravity (approximately 9.81 m/s²)
• $h$ is the height of the object above the reference point (m)

2.2.2 Elastic Potential Energy

Elastic potential energy is the energy stored in a deformable object, such as a spring, when it is stretched or compressed.

The formula for elastic potential energy is:

$PE = \frac{1}{2}kx^2$

where:

• $PE$ is the elastic potential energy (Joules, J)
• $k$ is the spring constant (N/m)
• $x$ is the displacement of the spring from its equilibrium position (m)

3. Conservation of Energy

The principle of conservation of energy states that the total energy of an isolated system remains constant. Energy can be transformed from one form to another, but it cannot be created or destroyed.

In a system where only conservative forces (like gravity and spring forces) are acting, the total mechanical energy (sum of kinetic and potential energy) remains constant.

$E_{total} = KE + PE = constant$

4. Work Done

Work is done when a force causes displacement.

The formula for work done by a constant force is:

$W = Fd \cos{\theta}$

where:

• $W$ is the work done (Joules, J)
• $F$ is the magnitude of the force (Newtons, N)
• $d$ is the magnitude of the displacement (m)
• $\theta$ is the angle between the force and the displacement

If the force and displacement are in the same direction, $\theta = 0^\circ$, and $\cos{0^\circ} = 1$, so $W = Fd$.

5. Power

Power is the rate at which work is done.

The formula for power is:

$P = \frac{W}{t}$

where:

• $P$ is the power (Watts, W)
• $W$ is the work done (Joules, J)
• $t$ is the time taken to do the work (seconds, s)

Alternatively, power can be expressed in terms of force and velocity:

$P = Fv$

where:

• $P$ is the power (Watts, W)
• $F$ is the magnitude of the force (Newtons, N)
• $v$ is the velocity (m/s)

6. Example Problem

A 2 kg block is released from a height of 10 m above the ground. Assuming no energy is lost to friction, calculate the velocity of the block just before it hits the ground.

Solution:

1. Initial potential energy: $PE_i = mgh = 2 \times 9.81 \times 10 = 196.2 J$
2. Final kinetic energy: $KE_f = \frac{1}{2}mv^2$
3. By the conservation of energy, $PE_i = KE_f$
4. $196.2 = \frac{1}{2} \times 2 \times v^2$
5. $196.2 = v^2$
6. $v = \sqrt{196.2} \approx 14.00 m/s$