Mechanical Energy, Grade 10 Physics
1/ Mechanical Energy

Important: Mechanical energy is the sum of the gravitational potential energy and the kinetic energy.
Mechanical energy, U , is simply the sum of gravitational potential energy (PE) and the kinetic energy (KE). Mechanical energy is defined as:

U = PE + KE (4.7)

U = mgh + \[\dfrac{1}{2}\]mv² (4.8)​

​

2/ Conservation of Mechanical Energy
The Law of Conservation of Energy states:
Energy cannot be created or destroyed, but is merely changed from one form into another.

Definition: Conservation of Energy
The Law of Conservation of Energy: Energy cannot be created or destroyed, but is merely changed from one form into another.

So far we have looked at two types of energy: gravitational potential energy and kinetic energy.

The sum of the gravitational potential energy and kinetic energy is called the mechanical energy. In a closed system, one where there are no external forces acting, the mechanical energy will remain constant. In other words, it will not change (become more or less). This is called the Law of Conservation of Mechanical Energy and it states:

The total amount of mechanical energy in a closed system remains constant.

Definition: Conservation of Mechanical Energy
Law of Conservation of Mechanical Energy: The total amount of mechanical energy in a closed system remains constant.

This means that potential energy can become kinetic energy, or vise versa, but energy cannot ’dissappear’. The mechanical energy of an object moving in the Earth’s gravitational field (or accelerating as a result of gravity) is constant or conserved, unless external forces, like air resistance, acts on the object.

We can now use the conservation of mechanical energy to calculate the velocity of a body in freefall and show that the velocity is independent of mass.

Important: In problems involving the use of conservation of energy, the path taken by the object can be ignored. The only important quantities are the object’s velocity (which gives its kinetic energy) and height above the reference point (which gives its gravitational potential energy).

Important: In the absence of friction, mechanical energy is conserved and

U$_{before}$ = U$_{after}$​

In the presence of friction, mechanical energy is not conserved. The mechanical energy lost is equal to the work done against friction.

∆U = U$_{before}$ − U$_{after}$ = work done against friction​

In general mechanical energy is conserved in the absence of external forces. Examples of external forces are: applied forces, frictional forces, air resistance, tension, normal forces. In the presence of internal forces like the force due to gravity or the force in a spring, mechanical energy is conserved.

3/ Using the Law of Conservation of Energy
Mechanical energy is conserved (in the absence of friction). Therefore we can say that the sum of the P E and the KE anywhere during the motion must be equal to the sum of the P E and the KE anywhere else in the motion.

We can now apply this to the example of the suitcase on the cupboard. Consider the mechanical energy of the suitcase at the top and at the bottom. We can say:


The suitcase will strike the ground with a velocity of 6,26 m·s−1.

From this we see that when an object is lifted, like the suitcase in our example, it gains potential energy. As it falls back to the ground, it will lose this potential energy, but gain kinetic energy. We know that energy cannot be created or destroyed, but only changed from one form into another. In our example, the potential energy that the suitcase loses is changed to kinetic energy.

The suitcase will have maximum potential energy at the top of the cupboard and maximum kinetic energy at the bottom of the cupboard. Halfway down it will have half kinetic energy and half potential energy. As it moves down, the potential energy will be converted (changed) into kinetic energy until all the potential energy is gone and only kinetic energy is left. The 19,6 J of potential energy at the top will become 19,6 J of kinetic energy at the bottom.

Worked Example 1: Using the Law of Conservation of Mechanical Energy
Question: During a flood a tree truck of mass 100 kg falls down a waterfall. The waterfall is 5 m high. If air resistance is ignored, calculate

1. the potential energy of the tree trunk at the top of the waterfall.
2. the kinetic energy of the tree trunk at the bottom of the waterfall.
3. the magnitude of the velocity of the tree trunk at the bottom of the waterfall.

Answer
Step 1 : Analyse the question to determine what information is provided

• The mass of the tree trunk m = 100 kg
• The height of the waterfall h = 5 m.

These are all in SI units so we do not have to convert.
Step 2 : Analyse the question to determine what is being asked

• Potential energy at the top
• Kinetic energy at the bottom
• Velocity at the bottom

Step 3 : Calculate the potential energy.
P E = mgh = (100)(9,8)(5) = 4900 J
Step 4 : Calculate the kinetic energy.
The kinetic energy of the tree trunk at the bottom of the waterfall is equal to the
potential energy it had at the top of the waterfall. Therefore KE = 4900 J.
Step 5 : Calculate the velocity.
To calculate the velocity of the tree trunk we need to use the equation for kinetic energy.


Worked Example 2: Pendulum
Question: A 2 kg metal ball is suspended from a rope. If it is released from point A and swings down to the point B (the bottom of its arc):

1. Show that the velocity of the ball is independent of it mass.
2. Calculate the velocity of the ball at point B.

Answer
Step 1 : Analyse the question to determine what information is provided

• The mass of the metal ball is m = 2 kg
• The change in height going from point A to point B is h = 0,5 m
• The ball is released from point A so the velocity at point, vA = 0 m·s$^{−1}$.

All quantities are in SI units.
Step 2 : Analyse the question to determine what is being asked

• Prove that the velocity is independent of mass.
• Find the velocity of the metal ball at point B.

Step 3 : Apply the Law of Conservation of Mechanical Energy to the situation
As there is no friction, mechanical energy is conserved. Therefore:

As the mass of the ball m appears on both sides of the equation, it can be eliminated so that the equation becomes:

gh$_{A}$ = \[\dfrac{1}{2}\](v$_{B}$)² => 2gh$_{A}$ = v$_{B}$²​

This proves that the velocity of the ball is independent of its mass. It does not matter what its mass is, it will always have the same velocity when it falls through this height.
Step 4 : Calculate the velocity of the ball
We can use the equation above, or do the calculation from ’first principles’:

2gh$_{A}$ = v$_{B}$² => v$_{B}$ = \[\sqrt{9,8}\] m · s$^{−1}$​

Exercise: Potential Energy
E - 1. A tennis ball, of mass 120 g, is dropped from a height of 5 m. Ignore air friction.
(a) What is the potential energy of the ball when it has fallen 3 m?
(b) What is the velocity of the ball when it hits the ground?
E - 2. A bullet, mass 50 g, is shot vertically up in the air with a muzzle velocity of 200m·s$^{−1}$. Use the Principle of Conservation of Mechanical Energy to determine the height that the bullet will reach. Ignore air friction.
E - 3. A skier, mass 50 kg, is at the top of a 6,4 m ski slope.
(a) Determine the maximum velocity that she can reach when she skies to the bottom of the slope.
(b) Do you think that she will reach this velocity? Why/Why not?
E - 4. A pendulum bob of mass 1,5 kg, swings from a height A to the bottom of its arc at B. The velocity of the bob at B is 4 m·s$^{−1}$. Calculate the height A from which the bob was released. Ignore the effects of air friction.
E - 5. Prove that the velocity of an object, in free fall, in a closed system, is independent of its mass.

Xem thêm:
Physics 10.III Gravity and Mechanical Energy

High School Students Studying the Sciences Physics