Lab 4 - Conservation of Mechanical Energy

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

E

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

K

and potential energy

U

of the system.

( 1 )

E = K + U

In the absence of non-conservative 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

,

U₂

are the kinetic and potential energies at two different locations 1 and 2 respectively, then the 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

K

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

m

and velocity or speed

v

of the object by

( 4 )

K_T =

1

2

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

I

and angular velocity

ω

of the object by

( 5 )

K_R =

1

2

I ω²

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

R

of the sphere.

( 6 )

ω =

v

R

Just as the mass

m

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

I

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 =

2

5

MR²

where

M

and

R

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

U

. 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

g

is the acceleration due to gravity,

m

is the mass of the object, and

h

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 Eq. (2)

K₁ + U₁ = K₂ + U₂

.

Objective

The objective of this experiment is to determine if the law of conservation of mechanical energy can predict the velocity of a rolling sphere after rolling down a ramp. You will test your hypothesis by placing the sphere at various heights and measuring the velocity at the end of the ramp. You will compare the measured velocity with the predicted velocity using the conservation of mechanical energy. If we let the sphere roll down a ramp, its vertical height will decrease while its translational and rotational velocity will increase. Potential energy is lost and kinetic energy is gained. Assuming that the sphere rolls without slipping and there is no friction or air drag, the loss of potential energy will equal the gain in kinetic energy. Based on conservation of mechanical energy, if the sphere is placed at height

h₁

, as shown in Fig. 1, you can show that the final speed

v₂

at the end of the ramp is

( 10 )

v₂ =

10gh1 7

Figure 1: Sphere rolling down a ramp

Equation (10)

v₂ =

10gh1 7

tells us that we can predict the final translational speed

v₂

of the sphere at point at the bottom of the inclined portion of the ramp if we know the initial height

h₁

. In the experiment you will measure the translational speed of the sphere at the bottom of the ramp and compare it to the speed predicted by the conservation of mechanical energy. If these two (experimental and predicted) values of the translational speed are in close agreement, you can say that you have verified the law of conservation of mechanical energy.

Experimental Design

Let us consider several ways in which the translational speed of a small sphere can be measured.

• 1
  Use a police radar speed gun: A radar gun or speed gun is a small Doppler radar unit used to detect the speed of objects, especially trucks and automobiles for the purpose of regulating traffic, as well as pitched baseballs, runners or other moving objects in sports. It relies on the Doppler Effect applied to a radar beam to measure the speed of objects at which it is pointed. Radar guns may be hand-held or vehicle-mounted; see "Radar gun" or "Police RADAR". A radar speed gun measures the instantaneous speed of the sphere. As the sphere rolls down the ramp, its speed changes. It would be difficult to read the radar gun at the precise moment when the small sphere reaches the bottom of the ramp.
• 2
  Use an ultra sonic motion detector: The sonic motion detector transmits a burst of ultrasonic pulses. The ultrasonic pulses reflect off a target and return to the face of the sensor. The target indicator flashes when the transducer detects an echo. The sensor measures the time between the trigger rising edge and the echo rising edge. It uses this time and the speed of sound to calculate the distance to the object. To determine velocity, it uses consecutive position measurements to calculate the rate of change of position. See, for example, PASCO or Vernier. In general, a sonic motion detector used in a physics lab can only detect objects no smaller than a baseball. Measuring the speed of a small steel marble would be unreliable.
• 3
  Use a photogate: A photogate monitors the motion of objects passing through its gate, counting events as the object breaks the infrared beam. If the size of the object is known, by measuring the time an object blocks the gate, you can determine the velocity. See, for example, Vernier or PASCO. The size of the infrared beam from a photogate is not small compared to the diameter of the sphere. Thus you could not make an accurate determination of the length passing through the infrared beam.
• 4
  Use kinematics: Once the sphere leaves the ramp, it is under free fall. By measuring the vertical and horizontal components of the sphere's displacement after it leaves the ramp, you can determine the sphere's speed at the end of the ramp. Measuring the horizontal and vertical displacements of the sphere does not involve the use of motion detectors or photogates and is straight forward. You will use this method for the experiment.

Finding the speed using kinematics

The ramp has a short horizontal section from point to point . We assume that the sphere's speed down the ramp does not change when it travels on the short horizontal section between point and point . Once the sphere leaves the ramp, there is no force acting on it to change its horizontal velocity (assuming there is no air resistance). Gravity pulls the sphere down the instant it leaves the ramp. We will consider the equations of motion for the vertical and horizontal motions of the sphere. The vertical distance

h₂

that the sphere falls in time

ΔT

is given by

( 11 )

h₂ =

1

2

g(ΔT)²

where

g

= 9.81 m/s² is the acceleration due to gravity. The initial vertical component of the velocity of the sphere as it leaves the ramp is zero. The horizontal distance

d

the sphere moves from the end of the ramp to the point where it touches the table is given by

( 12 )

d = v₂(ΔT)

Using Eq. (11) and (12) we can show that

( 13 )

v_2kinematics = d

g 2h2

Equipment

Procedure

You will place a small sphere at 5 different vertical heights

h₁

, allow it to roll down the ramp and read the distance

d

from the edge of the ramp to the location where it hits the table. Independently, you will make a graph of the horizontal velocity

v₂

versus horizontal distance

d

using Eq. (13)

v_2kinematics = d

g 2h2

. From this graph you can read the horizontal velocity knowing the horizontal distance the sphere traveled for the 5 different vertical heights. From your measurements of the horizontal and vertical distances that the sphere travels after leaving the end of the ramp, you will determine the experimental velocity for the 5 different vertical heights using Eq. (13)

v_2kinematics = d

g 2h2

and check these values with your graph. The apparatus consists of an inclined ramp whose height you can vary by means of the adjustable clamp, as shown in Figs. 2 and 3.

Setting up

Procedure A: Determining the experimental velocity

v_2exp

using kinematics

Procedure B: Using the Conservation of Mechanical Energy to predict the velocity

v_2CME

Copyright © 2010 Advanced Instructional Systems, Inc. and North Carolina State University. | Credits