Conservation of mechanical energy

I. In the previous chapter it was noted that energy as well as momentum obeys the conservation law. Let us consider the conditions for energy conservation on the example of two bodies interacting with each other only. Such bodies are treated as an isolated system.

In the isolated system the interacting bodies can possess both energies kinetic and potential at the same time. For example, the artificial satellite of the Earth has kinetic energy due to its motion.

Besides, the artificial satellite - Earth system possesses potential energy because the satellite and the Earth interact with each other by a force of universal gravitation. Colliding balls possess at the

Fig. 71 same time kinetic energy due to their motion and potential energy

due to their elastic deformation.

II. The sum of a kinetic and potential energy of the system of bodies interacting with each other is called the total mechanical energy of the system:

For example, let a body with mass m to be at height h₁ above the ground and to have velocity (fig.71). At this state the body possesses kinetic energy and a stored potential energy . Then the total mechanical energy of the system is equal to .

III. Now we assume that the body descends at height h₂ where its velocity is . In this motion the force of gravity does work . Due to this work done the body gains kinetic energy:

As the left sides of these expressions are equal then the right sides are equal too, i.e.

Or this equation can be rewritten as

Hence, the change in kinetic energy is equal to the change in potential energy but they have opposite signs:

This equation says that if the potential energy of bodies increases on a certain amount then their kinetic energy decreases on the same amount and vice versa. Hence, we can conclude that one form of energy is converted into another form of energy. The above expression we rewrite as:

This expression states that the sum of kinetic and potential energy of bodies in an isolated system where they interact with each other by the force of gravity and elastic force is always constant. This is the law of energy conservation.

As the sum of kinetic and potential energy of a system of bodies gives the total mechanical energy then this law can be formulated as: the total mechanical energy of an isolated system of bodies interacting with each other by the force of gravity and elastic force is conserved

In a more general form the law of energy conservation can be formulated as: in nature the energy is never created or destroyed; it is only transferred from one form to another.

Conversion of potential energy to kinetic or kinetic to potential is one of the remarkable phenomena in nature. It is the basic distinctive feature of energy.

The law of conservation and conversion of energy allows better understanding the physical sense of work. Since one the same work is done to increase the kinetic energy as well as to decrease the potential energy it follows that the work is equal to energy converted from one form to another.

The law of conservation of a total mechanical energy is used for solution of many problems of mechanics.

Self-testing questions

1. What is the total mechanical energy?

2. Is there a conversion of kinetic energy to potential energy during oscillation of a pendulum of a wall clock? Explain how it occurs.

3. How does your energy change when you raise up a book, throw a ball, slide a chair across the floor?

4. What does the law of conservation of total mechanical energy state?

5. How to explain the physical sense of work using the law of conservation and conversion of energy?

Exercise 20

1. Calculate the maximum height of the arrow fired from the bow vertically upward. Initial speed of the arrow is 40 m/s. You can neglect air resistance.

2. The body with mass 2 kg falls from the height of 30 m above the ground. Calculate the kinetic energy of the body at the height of 15 m above the ground and just before it strikes the earth.

3. The shell is fired from the gun vertically upward with initial speed of 280 m/s. At what height above a firing point its kinetic energy is equal to its potential energy?

4. A steam hammerhead falling from height of 8 m above the ground just before it hits the ground possesses the kinetic energy of 18 000 J. What is the mass of the steam hammerhead?

5. The body with mass 400 g is attached to the compressed spring with force constant of 100 N/m. When the spring is released the body oscillates with maximum elongation of the spring of 10 cm. What is the maximum speed of the oscillating body? The mass of the spring can be neglected.

Date: 2015-01-12; view: 1115