Work and Energy notes

MDCAT Physics

This chapter explains work done by a force, kinetic and potential energy, power, the work-energy theorem, and the conservation of energy. It also covers frictional losses, efficiency, electrical energy units, rotational kinetic energy, and practical energy calculations.

Work Done by a Force

Work is done when a force produces displacement in its direction or against its direction. Work is a scalar quantity, so it has magnitude but no direction.

If a constant force F acts on a body and produces displacement d, the work done is W = Fd cos θ, where θ is the angle between force and displacement. The SI unit of work is joule (J).

Work is the product of the component of force along displacement and displacement: W = Fd cos θ.
If θ = 0°, W = Fd, so work is maximum and positive.
If θ = 90°, W = 0 because the force has no component along displacement.
If θ = 180°, W = -Fd. The work is negative because force and displacement are opposite.
For F = 20 N, d = 10 m, and W = 100 J, cos θ = 100/(20 × 10) = 0.5, so θ = 60°.
The work done by centripetal force is zero because centripetal force is perpendicular to instantaneous displacement.
Work done by brakes on a moving vehicle is negative because braking force acts opposite to motion.

Work in Special Situations

Different forces can do positive, negative, or zero work. The sign of work depends on the angle between the force and displacement, not simply on whether a force is present.

On an inclined plane, the component of gravitational force parallel to the plane is responsible for motion down the plane. If the plane makes angle θ with the horizontal, this component is mg sin θ.

The component of weight parallel to an inclined plane is mg sin θ.
The component of weight perpendicular to an inclined plane is mg cos θ.
The normal reaction is perpendicular to motion along a smooth inclined plane, so its work is zero.
Friction usually does negative work because it opposes relative motion.
A force can do zero work even when it acts on a moving body, as in uniform circular motion by centripetal force.
The work done by a variable force is equal to the area under the force-displacement graph.
When an object is lifted at constant speed, the external force does positive work and gravity does equal negative work.

Kinetic Energy

Kinetic energy is the energy possessed by a body due to its motion. For a body of mass m moving with speed v, kinetic energy is KE = 1/2 mv^2.

Kinetic energy is a scalar quantity. It is always zero or positive. Since it depends on v squared, doubling speed makes kinetic energy four times greater.

The SI unit of kinetic energy is joule.
KE = 1/2 mv^2.
Momentum is p = mv, so KE = p^2/(2m) and also KE = 1/2 pv.
For the same mass, kinetic energy is proportional to the square of speed.
If kinetic energy is quadrupled and mass remains constant, momentum is doubled because p is proportional to the square root of KE.
The work required to stop a moving object is equal to its initial kinetic energy in magnitude.
A moving car at twice the speed has four times the kinetic energy and requires four times the stopping distance under the same braking conditions.
For rotational motion, kinetic energy is KErot = 1/2 Iω^2, where I is moment of inertia and ω is angular speed.

Potential Energy and Gravitational Energy

Potential energy is stored energy due to position, shape, or configuration. Near the Earth's surface, gravitational potential energy is measured relative to a chosen reference level.

When a body of mass m is raised through height h, its gravitational potential energy increases by PE = mgh. The value of potential energy depends on the selected reference level, but the change in potential energy does not.

Gravitational potential energy near Earth is PE = mgh.
g is approximately 9.8 m/s^2, although 10 m/s^2 may be used when stated or allowed.
Work done in lifting a body slowly through height h is mgh.
Work done by gravity when a body falls through height h is positive and equal to mgh.
Work done against gravity in raising a body is positive, while work done by gravity is negative during upward motion.
Elastic potential energy of a stretched or compressed spring is PE = 1/2 kx^2.
The zero level of gravitational potential energy can be selected conveniently.

Conservative and Non-Conservative Forces

A conservative force is a force for which work done depends only on the initial and final positions, not on the path followed. The total work done by a conservative force around a closed path is zero.

A non-conservative force depends on the path followed. It changes mechanical energy into other forms, such as heat and sound.

Gravity is a conservative force.
Electrostatic force is a conservative force.
Spring force is a conservative force within its elastic limit.
Friction is non-conservative because its work depends on the distance travelled.
Air resistance is non-conservative.
For a conservative force, work done equals the negative change in potential energy: W = -ΔPE.
For a non-conservative force, mechanical energy is not conserved by itself.
In the presence of friction, total energy is still conserved, but some mechanical energy changes into thermal energy.

Work-Energy Theorem and Conservation of Energy

The work-energy theorem states that the net work done on a body is equal to the change in its kinetic energy. Thus, Wnet = ΔKE = KEfinal - KEinitial.

This theorem is useful when force and displacement are known, or when initial and final speeds are given. It applies to both constant and variable forces.

Net work done on a particle is equal to its change in kinetic energy.
If net work is positive, kinetic energy and speed increase.
If net work is negative, kinetic energy and speed decrease.
If net work is zero, kinetic energy and speed remain constant.
The work needed to stop an object is equal in magnitude to its initial kinetic energy.
For a freely falling body without air resistance, loss of gravitational potential energy equals gain in kinetic energy.
In an ideal system, total mechanical energy remains constant: KE + PE = constant.
For a body dropped from height h, just before reaching the ground, mgh is converted into kinetic energy, so v = √(2gh), if air resistance is neglected.

Power and Its Units

Power is the rate at which work is done or energy is transferred. Average power is P = W/t. Instantaneous power is P = Fv when force and velocity are in the same direction.

If a force F makes angle θ with velocity v, power is P = Fv cos θ. The SI unit of power is watt.

One watt is one joule per second: 1 W = 1 J/s.
Average power is P = W/t.
For a body moving with velocity v under force F, P = Fv cos θ.
If a person pulls a block with tension T at angle θ while the block moves with speed V, power delivered is P = TV cos θ.
One horsepower is equal to 746 W.
One megawatt hour is 3.6 × 10^9 J, which is also 3.6 GJ.
Electrical energy consumed in domestic billing is commonly measured in kilowatt-hours. One unit of electricity is 1 kWh.
An appliance rated 500 W used for 24 hours consumes 500 × 24 = 12,000 Wh = 12 kWh, or 12 units.

Efficiency and Energy Losses

In practical machines, the input energy is greater than the useful output energy because some energy is lost as heat, sound, vibration, or deformation. Efficiency compares useful output with total input.

Efficiency is expressed as a fraction or percentage. It cannot be greater than 100 percent for a real machine.

Efficiency = useful output energy/input energy.
Percentage efficiency = (useful output energy/input energy) × 100.
The same formula can be written using power: efficiency = useful output power/input power.
Friction changes mechanical energy into thermal energy.
For a pump lifting water, useful output energy is mgh and input energy is Pt.
If an engine has input power 3.3 kW and efficiency 60 percent, useful power is 0.60 × 3.3 kW = 1.98 kW.
A 3.3 kW pump operating for 5 s at 60 percent efficiency supplies 9,900 J of useful energy. From mgh = 9,900 J, it can lift 100 kg through 10 m.
Energy is not destroyed in losses. It is transformed into less useful forms.

Applications and Numerical Results

Work and energy principles explain stopping distances, heating, lifting, pumping, and motion on inclined planes. Always convert quantities into SI units before using formulas.

Stopping distance depends strongly on speed. If braking force is constant, stopping distance is proportional to the square of speed.

If a car stops in 6 m at 50 km/h, its stopping distance at 100 km/h is 6 × (100/50)^2 = 24 m.
For heating, energy supplied is Q = mcΔT, where c is specific heat capacity.
Heating 500 g of water from 20°C to 50°C requires Q = 500 × 4.2 × 30 = 63,000 J.
If this heating takes 5 minutes, heater power is P = 63,000/300 = 210 W.
Lifting 20 kg through 2 m in 4.9 s requires power P = mgh/t = (20 × 9.8 × 2)/4.9 = 80 W.
For a ball falling through height h and then penetrating sand by x, the standard approximation for average resistive force is R = mgh/x.
If the downward work of the ball's weight during penetration is also included exactly, R x = mg(h + x), giving R = mg(h + x)/x.
Geothermal energy comes from the Earth's internal heat, including heat from radioactive decay and residual heat within the Earth.

Escape Velocity and Rotational Energy

Escape velocity is the minimum speed needed for an object to escape a planet's gravitational field without further propulsion. It depends on the planet's mass and radius.

Rotational kinetic energy depends on both angular speed and the distribution of mass about the axis. Objects with different shapes can have different rotational kinetic energies even when their masses and radii are equal.

Escape velocity is ve = √(2GM/R), where G is the gravitational constant, M is planetary mass, and R is planetary radius.
Escape velocity does not depend on the mass of the escaping object when air resistance is ignored.
Among the planets of the Solar System, Jupiter has the greatest escape velocity.
For rotation, KErot = 1/2 Iω^2.
For a thin hoop about its central axis, I = MR^2.
For a uniform disc about its central axis, I = 1/2 MR^2.
For the same mass, radius, and angular speed, a hoop has twice the rotational kinetic energy of a disc.
The hoop therefore has KEhoop = 2 KEdisc.

Key terms

Work: Work is the product of force and displacement in the direction of the force, W = Fd cos θ.
Joule: A joule is the SI unit of work and energy, equal to one newton metre.
Kinetic energy: Kinetic energy is the energy possessed by a body because of its motion, KE = 1/2 mv^2.
Potential energy: Potential energy is stored energy due to position, shape, or configuration.
Gravitational potential energy: Gravitational potential energy near Earth's surface is mgh relative to a selected reference level.
Power: Power is the rate of doing work or transferring energy, P = W/t.
Watt: A watt is the SI unit of power and equals one joule per second.
Work-energy theorem: The net work done on a body equals the change in its kinetic energy.
Conservative force: A force whose work depends only on initial and final positions is conservative.
Non-conservative force: A force whose work depends on the path followed is non-conservative.
Efficiency: Efficiency is the ratio of useful output energy or power to input energy or power.
Escape velocity: Escape velocity is the minimum speed required to escape a planet's gravitational field.
Specific heat capacity: Specific heat capacity is the heat required to raise the temperature of unit mass of a substance by one degree Celsius.
Rotational kinetic energy: Rotational kinetic energy is the energy of rotation, given by 1/2 Iω^2.
Stopping distance: Stopping distance is the distance travelled by a moving body from application of brakes until it stops.

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Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

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