Conservation of Mechanical Energy – Concepts

Introduction

When a body moves, some things—such as its position, velocity, and momentum—change. It is interesting and useful to consider things that do not change. The total energy is a quantity that does not change; we say that it is conserved during the motion. There are several forms of energy with which you may be familiar, such as solar, nuclear, electrical, and thermal energies. In this experiment, you will investigate two kinds of mechanical energy: kinetic energy and potential energy. You will carry out an experiment that demonstrates the conservation of the total mechanical energy of a system.

Discussion of Principles

Conservation of Mechanical Energy

The total mechanical energy

of a system is defined as the sum of the kinetic energy

and potential energy

of the system.

( 1 )

E = K + U

In the absence of nonconservative forces, such as friction or air drag, the total mechanical energy remains a constant and we say that mechanical energy is conserved. If

K₁,

U₁,

K₂,

and

U₂

are the kinetic and potential energies at two different locations 1 and 2 respectively, then conservation of mechanical energy leads to the following mathematical expression.

( 2 )

K₁ + U₁ = K₂ + U₂

This can also be expressed in terms of the changes in kinetic and potential energies

( 3 )

ΔK + ΔU = 0

where

ΔK = K₂ − K₁

and

ΔU = U₂ − U₁.

Conservation of Mechanical Energy is one of the fundamental laws of physics that is also a very powerful tool for solving complex problems in mechanics.

Kinetic Energy

Kinetic energy

is the energy a body has because it is in motion. When work is done on an object, the result is a change in the kinetic energy of the object. Energy of motion can be translational kinetic energy and/or rotational kinetic energy. Translational kinetic energy

K_T

is the energy an object has because it is moving from place to place, regardless of whether it is also rotating;

K_T

is related to the mass

and velocity or speed

of the object by

( 4 )

K_T =

mv².

Rotational kinetic energy

K_R

is the energy an object has because it is rotating, regardless of whether the body as a whole is moving from place to place.

K_R

is related to the moment of inertia,

and angular velocity,

ω,

of the object by

( 5 )

K_R =

Iω².

The angular velocity of a rolling sphere that is not slipping is the velocity (relative to the center of the sphere) of a point on the circumference, divided by the radius

of the sphere.

( 6 )

ω =

Just as the mass

of a body is a measure of its resistance to a change in its (translational) velocity, the moment of inertia

of an object is a measure of that object's resistance to a change in its angular velocity. The moment of inertia of a solid sphere (of uniform density) is given by

( 7 )

I =

MR²

where

and

are the mass and radius of the sphere, respectively. The total kinetic energy

K_total

of the rolling sphere used in this experiment is the sum of the translational and rotational kinetic energies.

( 8 )

K_total = K_T + K_R

Potential Energy

An object might have energy by virtue of its position on account of the work done to put it there. The object is said to have potential energy

Gravitational potential energy, which we will be concerned with in this experiment, depends on the mass of the object, the acceleration due to gravity, and its location. It is important to remember that potential energy is only defined relative to a location, which can be chosen arbitrarily (that is, for convenience) without affecting the body's subsequent motion. For example, if the rolling sphere is held 1/2 meter above a table top, and the table top is 1 meter above the floor, the potential energy of the sphere has one value relative to the table top and a larger value relative to the floor. The potential energy of an object near the surface of the Earth is given by

( 9 )

U = mgh

where

is the acceleration due to gravity,

is the mass of the object, and

is the height above the chosen reference level (the end of the ramp, where the potential energy is zero by choice). If we drop an object, it will lose height and gain velocity as it falls in accordance with equation 2.