### Pendulums

A pendulum is a system where you have a mass at the end of a string, and the string itself is of negligible mass.
The string is attached to a fixed object, and the mass is allowed to swing from A to B to C, and then backwards, over and over. This is similar to the swings at the playground.

Diagram 1

Height h

The **height h** is the vertical distance from the release point to the lowest point that the mass can reach. It does not matter where the ground is. The height is not measured in terms of how high
above the ground the mass is held.

In this example the height is 3m.

Diagram 2

Gravitational Potential Energy E_{p}

When the mass is at A, it has maximum Gravitational Potential Energy E_{p}.

At A the mass is at rest so :

E_{k} = 0 joules.

Mechanical Energy

**It is important to note that at A, the mass has both potential and kinetic energies, even though the kinetic energy is zero. **

Mechanical Energy at A is:

E_{m} = E_{p} + E_{k}

E_{m} = 58,8 + 0

E_{m} = 58,8 J

When their are NO outside forces, such as friction, mechanical energy would be conserved.

**It is important to note that at B, the mass has both potential and kinetic energies, even though the potential energy is zero. **

Kinetic Energy E_{k}

At B the Em would also = 58,8J

At B the h = 0,

so E_{p} = 0

Thus all the E_{m} would become kinetic energy.

E_{k} at B would be 58,8J.

The speed at B can thus be calculated:

There are formal ways of writing all this down, but tends to appear complicated. We will practice that later.

Question 1

**A block of wood of mass 4kg is released from a vertical height of 2m with a pendulum action as shown.**

**1.1. Calculate the Ep at the highest point.**

**1.2. State the Ek of the block at the lowest point of it’s swing. State the law used.**

E_{k} = 78,4J

Law of Conservation of Mechanical Energy

**1.3 Calculate the speed of the block at the lowest point.**

Question 2

**A 4kg masspiece, when released from a height h, passes by it's lowest point of swing with a speed of 8m.s**^{-1}.

**2.1. Calculate the E**_{K} at the lowest point of the swing.

**2.2. What is the E**_{P} at the lowest point of the swing? Why.

The E_{P} = 0

The h = 0

**2.3. Calculate the mechanical energy.**

E_{m} = E_{p} + E_{k}

E_{m} = 0 + 128

E_{m} = 128 J

**2.4. State the maximum potential energy.**

max E_{P} = 128 J

It is important to state "maximum" since there are many heights and hence many potential energies.

This means "at the greatest height" reached.

This also means the speed at that height is zero.

**2.5. Calculate the height from which the mass was released.**

Very formal :

It is wise to get used to the simple way of solving, and then practise the formal layouts.