9709. Mechanics 1. Work-Energy & Power

To keep in mind throughout: Energy is a scalar property

Work-Energy Equation

If a constant force acts on an object over a certain distance, the work done (by all forces with a component in the direction of motion) is equal to the gain in kinetic energy of the object.

So, $\Sigma Fs = \frac{1}{2} mv^2 - \frac{1}{2} mu^2$

As with Newton’s Second Law, when we use the work-energy equation the left-hand side (LHS) should show net force, i.e. we must consider any resisting forces:

𝝨Fscos𝜽 – Rs = Δ_KE

Worked examples

A man uses a constant force of 250N to push a box of mass 20kg a distance of 4m in a curved path across a horizontal floor. The box starts from rest. Find the final speed of the box:
- When the floor is smooth;
- When the coefficient of friction between the floor and the box is 0.12
A bag of mass 0.05 kg is thrown vertically upwards with an initial speed of u ms^-1. It rises through a distance of 1.5m and then falls through 2.5m before hitting the floor. It hits the floor with speed v ms^-1. Throughout the motion, air resistance of 0.01N acts on the ball. Calculate the initial speed, u ms^-1 and the final speed v ms^-1.

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Conservation of Energy in System of Conservative forces

A conservative force is one whose work done in moving a particle between 2 points is independent of the path taken, e.g. weight.

By contrast, friction and a driving force are non-conservative forces.

Work done by a conservative force changes potential energy into kinetic energy with no loss of mechanical energy (e.g. boy sliding down a smooth hill).

For a closed system of conservative forces, total mechanical energy is constant, i.e. KE₀ + PE₀ = KE₁ + PE₁ (where subscript 1 can represent any point in the motion, e.g. the end point). This is called conservation of mechanical energy.

Worked Examples

A box of mass m kg is initially at rest. It slides down a smooth slope inclined at 30º to the horizontal. Find the speed of the box after sliding a distance of 3m.
A ball of mass 0.05kg is thrown vertically upwards from a height of 1.5m above the ground. The ball rises through a height of 2m to reach its maximum height at 3.5m above the ground. Use the conservation of mechanical energy to find the initial speed of the ball.

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Conservation of Energy in System of Non-Conservative forces

As stated above, a non-conservative force is something like friction, air resistance or driving force, for which the work done is different if a different path is taken.

Work done by a non-conservative force converts energy into movement (e.g. driving force converts chemical energy from fuel into kinetic energy). Total energy is conserved, but mechanical energy is increased. Of course, friction and air resistance decrease mechanical energy, dissipating it into non-mechanical energy (e.g. heat and sound energy).

Worked Examples

A ball of mass 50g falls from rest through a height of 80cm. It hits the ground and rebounds to a height of 30cm. Find the mechanical energy lost in the motion from the start at height 80cm to the end at height 30cm above the ground.
A crate of mass 50kg slides across a rough horizontal floor. The crate has an initial speed of 3 ms^-1 and is brought to rest by friction. The distance travelled by the crate is 4m. Find the coefficient of friction between the floor and the crate.
A parcel of mass 3kg slides 3.5m down a rough slope inclined at 20º to the horizontal. The coefficient of friction between the parcel and the slope is 0.5. When it reaches the bottom of the slope the parcel has speed 8ms^-1. Use the work-energy principle to find the speed of the parcel at the top of the slope.

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Power

Power is the rate of doing work. As work done is Fs, so we can consider the average power generated power as Fs/t, which is the same as Fv. Power = Fv. It is a scalar quantity measured in Watts.

For a specific amount of power, the driving force generated will be greater at lower speeds and smaller at higher speeds. The maximum power output of an engine can be used to find the maximum speed that it can generate.

Worked Examples

A car of mass 1500kg is being driven along a level road. It accelerates from 0kmh^-1 to 100kmh^-1 in 10s. Air resistance and friction may be ignored. Use the work-energy principle to calculate the average power generated by the engine.
A car of mass 1500kg has an engine that has a maximum power output of 200kW. The resistance to motion is typically 5000N.
- Find the maximum speed that the car can achieve on a level road (ignoring speed restrictions);
- Find the instantaneous acceleration of the car when the engine is working at its maximum power and the car is travelling at 20ms^-1;
- Find the instantaneous acceleration of the car when the engine is working at its maximum power and the car is travelling at 20ms^-1 up a hill that is inclined at sin^-10.28 to the horizontal.