Gravitational potential energy (GPE) is important for understanding how things move when gravity is involved.

GPE is the energy that an object has because of its position in a gravitational field.

To find out how much gravitational potential energy an object has, we need to consider three things:

1. The object's mass.
2. The height of the object.
3. The strength of gravity acting on it.

The formula to calculate gravitational potential energy looks like this:

$$U = mgh$$

Here’s what each letter means:

• ( U ) is the gravitational potential energy.
• ( m ) is the mass of the object.
• ( g ) is the acceleration due to gravity (which is about ( 9.81 , \text{m/s}^2 ) near the Earth’s surface).
• ( h ) is how high the object is above a reference point.

Let’s say we have three different weights:

• ( m_1 = 2 , \text{kg} )
• ( m_2 = 5 , \text{kg} )
• ( m_3 = 10 , \text{kg} )

If we lift them all to a height of ( h = 3 , \text{m} ), we can calculate their gravitational potential energy like this:

1. For the first object (( m_1 )):

   $$U_1 = m_1 g h = 2 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 3 \, \text{m} = 58.86 \, \text{J}$$

2. For the second object (( m_2 )):

   $$U_2 = m_2 g h = 5 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 3 \, \text{m} = 147.15 \, \text{J}$$

3. For the third object (( m_3 )):

   $$U_3 = m_3 g h = 10 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 3 \, \text{m} = 294.30 \, \text{J}$$

From these calculations, we can see that gravitational potential energy increases as the mass of the object increases when height and gravity stay the same.

It’s important to keep in mind how we measure height. We usually measure height from a starting point, like the ground. If we lift something higher, its gravitational potential energy will also increase.

For example, let’s imagine we have an object that is originally at ( 1 , \text{m} ) above the ground. If we lift it up to ( 4 , \text{m} ), we can find out how much the gravitational potential energy changes, using the weight ( m_2 = 5 , \text{kg} ):

1. First, we calculate the GPE when it's at ( 1 , \text{m} ):

   $$U_{initial} = m_2 g h_{initial} = 5 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 1 \, \text{m} = 49.05 \, \text{J}$$

2. Then, we calculate it at ( 4 , \text{m} ):

   $$U_{final} = m_2 g h_{final} = 5 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 4 \, \text{m} = 196.2 \, \text{J}$$

Finally, to find the change in gravitational potential energy from ( 1 , \text{m} ) to ( 4 , \text{m} ), we do this:

$$\Delta U = U_{final} - U_{initial} = 196.2 \, \text{J} - 49.05 \, \text{J} = 147.15 \, \text{J}$$

In summary, we can calculate gravitational potential energy for different weights using the formula ( U = mgh ). By changing the mass and height, we can see how energy changes in a gravitational field. This helps us understand how objects behave under gravity and sets the stage for more complex ideas in physics.