# How do conservative forces conserve mechanical energy?

A force that conserves mechanical energy is known as a conservative force.

Question: How do conservative forces conserve mechanical energy?

• The question of others may be, why haven't you tried Wikipedia or Google first ? Question without a priori effort are usually closed on StackExchange sites. – Poutnik May 24 at 20:12
• I have already searched on wikipedia but not able to get the answer. – Isha May 25 at 6:53
• Not even here ? – Poutnik May 25 at 7:08

## 1 Answer

It conserves it by insuring the total energy, potential plus kinetic energy, is constant.

The simplest example is the force of gravity. It is a conservative force. The total mechanical energy of a mass is the sum of its kinetic and potential energy. An object of mass $$m$$ sits on top of a table a height $$h$$ from the floor. The object has gravitational potential energy equal to $$mgh$$ with respect to the flooor (it's always with respect to something).

The object falls off the table. The gravitational field does work on the falling object converting its potential energy at the top of the table into kinetic energy as it falls. The sum of the potential energy and kinetic energy all the way down is constant with the potential energy decreasing and kinetic energy increasing by the same amount on the way down. Mechanical energy is conserved.

Just prior to impact with the floor all the object has is kinetic energy and no potential energy. The kinetic energy is then equal to its original potential energy, or

$$mgh=\frac{mv^2}{2}$$

Hope this helps.

• Technically, any force which can be described as the spatial gradient of a (scalar) potential is to be called conservative. If the potential is time-dependent, the total energy of a particle moving under this potential might not be conserved during the time-evolution. Correct me if I am mistaken. – Dvij Mankad May 24 at 20:46
• Agree with the first point. Applies to spring forces and electrostatic forces as well. Not sure I understand the second point. Can you give an example? – Bob D May 24 at 22:31
• Closed path force integral in time variable potential field has generally nonzero value. – Poutnik May 25 at 7:22
• Similarly as in the answer, planets, asteroids and comets, orbiting the Sun, keep their total mechanical energy $0.5mv^2 - \frac{GMm}{r}$ – Poutnik May 25 at 7:53