Lab 8 - 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

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 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,

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

of the object by

( 5 )

K_R =

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

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 Eq. (2).

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

Figure 1: Sphere rolling down a ramp

Equation (10) tells us that we can predict the final translational speed

v₂

of the sphere at point Point two

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 agree to within experimental uncertainties, 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 Point two

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 Point two

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 =

g(ΔT)²

where

= 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

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 Eq. (12) we can show that

( 13 )

v_2kinematics = d

2h₂

Equipment

Inclined ramp clamped to a table
Small steel sphere
Meter stick
Carbon paper
Bubble level

Procedure

You will place a small sphere at 5 different vertical heights

h₁,

allow it to roll down the ramp and read the distance

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

using Eq. (13). 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) and check these values with your graph.

Figure 2: Schematic diagram of the apparatus.

Figure 3: Actual apparatus.

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

Tape several pieces of notebook paper to the top of the table, extending out from the edge of the horizontal section of the ramp to about 1 m. Set the ramp so that it rests on the paper, so that

h₂ = 0.

Mark the position of the lower edge of the ramp on the paper.

Adjust the ramp so that the horizontal section is about 18 to 20 cm above the table. The ramp should be adjusted so that the horizontal section of the ramp is level, as indicated, by placing a small bubble level there. If the lower edge of the ramp is not level, ask your lab instructor for help with this adjustment. This will be position

h₂

of the ramp throughout the experiment. Make sure that you do not disturb it while doing the experiment.

Procedure A: Determining the Experimental Velocity
v_2exp
using Kinematics

Using Eq. (13) prepare a graph, in Excel, of the velocity

v_2kinematics

versus the horizontal distance

for two values of

(namely zero and 50 cm). You will use this graph to read the experimental velocity of the ball

v_2exp.

See Appendix G.

Determine and record the uncertainty in the horizontal distance measurements.

Calculate the uncertainty in the velocity

v_2kinematics

based on the uncertainty in measuring d. See Appendix C .

Checkpoint 1:
Ask your TA to check your Excel graph of

v_2kinematics

vs.

Mark 5 positions along the length of the ramp at intervals of about 2.5 cm. These are the positions from which you will release the sphere. Do not start too close to the bottom of the incline. Make sure these positions are evenly spaced.

Measure the height of these positions from the table top and determine the height

h₁

for each of these positions. Record them in the data table. This is the vertical distance that the sphere falls while rolling down the ramp.

Place a piece of carbon paper face down on top of the paper on the table. Do not tape the carbon paper to the table or to the white sheet of paper.

Place the sphere at the first position on the ramp, and release it. Note: You should place the center of the sphere at this mark and release it! Be careful not to push it down the ramp.

Remove the carbon paper. You will see a small spot on the underlying paper marking the point where the sphere landed. Measure and record the horizontal distance

from the line drawn earlier marking the end of the ramp to the point of impact. Once you have measured the position of the dot, draw an "X" through it, and label it so that you will be able to distinguish it from later marks. Record this distance in Data Table 1 (a) on the worksheet.

You will do three trials for the same value of

h₁

to get the average value of d. Therefore, repeat the process two more times for the same release position of the sphere, record the horizontal distance again, and then take the average of the three values of d. Record these values in Data Table 1 (a).

Repeat steps 9 through 11 for each value of

h₁

by releasing the sphere from the positions you marked on the incline. You might have to move the carbon paper out farther as the point of release goes further up the ramp. Label the impact points as they are produced so you do not get confused with marks made later. Note: For each value of

h₁

you should have three trials from which you will find the average value of the horizontal distance d.

Calculate the uncertainty in the average value of

for each

h₁

value and enter these values in Data Table 1 (b).

For each height

h₁,

use Eq. (13) to determine

v_2exp,

the velocity of the sphere as it leaves the ramp. Use the graph (see step 3) to verify your value of

v_2exp.

Using the uncertainty in

h₂

and

d_ave,

for each value of

v_2exp

calculate the uncertainty and the percent uncertainty. Enter these values in Data Table 1 (b). See Appendix B.

Checkpoint 2:
Ask your TA to check your calculations before proceeding.

Procedure B: Using the Conservation of Mechanical Energy to Predict the Velocity
v_2CME

Determine and record the uncertainty in the vertical measurement

h₁.

For each of the 5 values of

h₁

that you used in procedure A, calculate the horizontal velocity

v_2CME

using conservation of mechanical energy as given by Eq. (10), the uncertainty associated with it, and the percent uncertainty in these values. Enter these values in Data Table 2.

Compute the percent difference between

v_2exp

and

v_2CME

for each value of

h₁

and record these in Data Table 2.

For each value of

h₁,

use the percent uncertainties (last column in Data Table 1 (b) and Data Table 2) to compare the values of

v_exp

and

v_2CME.

Enter your results in Data Table 3.

Checkpoint 3:
Ask your TA to check your calculations.

Lab 8 - Conservation of Mechanical Energy

Introduction

Discussion of Principles

Conservation of Mechanical Energy

Kinetic Energy

Potential Energy

Objective

Experimental Design

Finding the speed using kinematics

Equipment

Procedure

Setting up

Procedure A: Determining the Experimental Velocity v2exp using Kinematics

Procedure B: Using the Conservation of Mechanical Energy to Predict the Velocity v2CME

Procedure A: Determining the Experimental Velocity
v_2exp
using Kinematics

Procedure B: Using the Conservation of Mechanical Energy to Predict the Velocity
v_2CME