Potential energy and its conservation
Stored energy
Water held high behind a dam can spin turbines when it is let go. A stretched bow can launch an arrow. Both store energy, ready to use.
Energy stored because of position or shape is called potential energy. We write it as V.
Lift a ball of mass m slowly through a height h. You pull up with a force mg, so you do work m g h.
That work is not lost. Let go, and gravity does exactly m g h of work as the ball falls back, turning it into kinetic energy.
So the gravitational potential energy at height h is V equals m g h.
Height from where? We choose a zero level, like the floor or the ground. Only changes in potential energy matter, so any choice works.
And the force is linked to V: F equals minus dV by dx. The force pushes the body towards lower potential energy.
Conservative forces and the conservation of energy
Carry a bag up to the third floor by the stairs, or by the lift. Either way, gravity does the same work: minus m g h.
When the work done by a force depends only on the start and end points, not the path, we call it a conservative force. Gravity and the spring force are conservative.
Take the bag up and bring it back down to where it started. Over the closed loop, gravity does zero work in total.
Friction is different. Drag a box around the room and back: friction does negative work the whole way. It is non-conservative.
Now drop a ball from height H. As it falls, potential energy drops and kinetic energy rises by exactly the same amount.
Watch the two bars: one shrinks, the other grows, and their total stays the same. K plus V is constant.
This is the conservation of mechanical energy: if only conservative forces do work, the total mechanical energy of a body stays the same.
At the ground, half m v squared equals m g H, so v equals root of 2 g H. Mass cancels out.
A bob on a string
A small bob of mass m hangs from a light string of length L. We give it a speed v₀ at the bottom so that it swings round in a vertical circle.
The tension in the string is always at right angles to the motion, so it does no work. Only gravity does work, so mechanical energy is conserved.
At the top, the bob is 2L higher than at the bottom. So half m v₀ squared equals half m v squared plus m g times 2L.
To stay on a circular path at the top, the bob needs enough speed. The least it can have is when gravity alone supplies the centripetal force: m g equals m v squared over L.
So at the top, v squared is at least g L. Put that into the energy equation.
The bob needs a speed of at least root 5 g L at the bottom to loop the loop. Any less and the string goes slack before the top.
Roller coasters use the same idea. The first hill must be high enough to carry the cars over every hump and loop that follows.
Try it: energy on a roller coaster
- Pick a release height and press Release. Watch KE and PE swap while their total stays flat.
- Without friction the cart rises to the same height on the far side. Turn friction on and see energy drain away as heat.
- Change the mass. Do the speeds change?
Quick check
Two carts, 2 kg and 8 kg, are released from rest at the same height on the same smooth track. At the bottom, which one is moving faster?
- The 8 kg cart
- The 2 kg cart
- Both have the same speed
- It depends on the shape of the track
Answer
C. Energy conservation gives mgh = ½mv², so v = √(2gh). The mass cancels, and the shape of the track does not matter on a smooth track. The 8 kg cart does have four times the kinetic energy, though.
Ball dropped from a building
A ball is dropped from rest from the top of a 20 m tall building. Ignoring air resistance, about how fast is it moving just before it hits the ground? (g = 9.8 m/s²)
- 14 m/s
- 20 m/s
- 196 m/s
- 28 m/s
Answer
B. All the potential energy becomes kinetic energy: mgh = ½mv², so v = √(2gh) = √(2 × 9.8 × 20) = √392 ≈ 19.8 m/s, about 20 m/s. 196 m/s is what you get if you forget the square root.
Summary
- Gravitational potential energy:
V = mgh - Force from potential energy:
F = −dV/dx - Conservation of mechanical energy:
K + V = constant - Speed after falling height h:
v = √(2gh)
- Potential energy is stored energy of position or shape. Only changes in V matter, so the zero level is our choice.
- A conservative force (gravity, spring) does work that depends only on the end points. Over a closed path it does zero work. Friction is non-conservative.
- If only conservative forces do work, K + V stays constant: kinetic and potential energy just swap.
- A bob on a string needs at least √(5gL) at the bottom to complete a vertical circle.