20F | PHY8103 – Physics 1

Lab 8 – Conservation of Mechanical Energy

Name:___________________________

Date:___________________________

REQUIRED READING

Class notes

OBJECTIVES OF THE LAB

You will develop an understanding of mechanical energy.

Become familiar with the notions of potential, kinetic and thermal energies.

Understand the difference between conservative and non-conservative systems.

You will see whether or not the Law of Conservation of Mechanical Energy applies to the cart on the ramp.

MATERIALS NEEDED

Your imagination

Ramp

Cart

1 Photo-Gate

Timer

Meter stick

SUMMARY OF THEORY

Momentum and energy are by far the most important physical variables used for the quantitative description of physical phenomenon. There are several reasons for this, in particular is the fact that for a closed, isolated physical system, the total energy and total momentum are always conserved. This was recognized by Newton and is contained in his Third Law.

When Newton first introduced the concept of momentum, he used the term “quantity of motion”, which conveys the significance of the associated quantities of mass and velocity. In this respect, mass and velocity can also be associated through kinetic energy.

We define kinetic energy, , by the following expression,

(8-1)

Let’s now consider another form of energy. Potential energy, , is a form of energy associated with a mass’ position relative to some reference point. In Lab 7 we saw how a spring stores elastic potential energy either through stretching or compressing. Another form of potential energy is gravitational potential energy. In order to visualize this, think of an object at some height, , above the ground. The object has stored potential energy as a function of its mass and height off the ground. We can then define gravitational potential energy by the following expression,

(8-2)

where is the gravitational acceleration.

If we want to now bring in the notion of energy conservation to an object suspended at some height off the ground, we can postulate that as the object falls it gains kinetic energy (through increased velocity) and loses potential energy (through loss in height).

We now define the total energy, , of the object to be the sum of the potential, , and kinetic energy, , as,

(8-3)

We can now assert that the total energy, , of the object is a conserved quantity. Particularly, , is a constant independent of time or position (and we will be ignoring friction and drag, ie thermal energy losses, but maybe not…).

(8-4)

To demonstrate how this works, consider an object located at rest at some height . It has a potential energy and a kinetic energy, . Therefore the total energy is (,

(8-5)

Next, the object is lifted by a distance above , its potential energy is now given by,

(8-6)

Suppose that now the object is released from the point and allowed to fall freely toward . What will its velocity be when it reaches ? At the total energy is given by (,

(8-7).

If we now want to determine an expression for the velocity of the object at point we can write the following expression (reminder: for a conservative system potential energy is transformed into kinetic energy):

(8-8)

So that,

(8-9)

And the velocity is then obtained from,

(8-10)

(8-11)

As can be seen from Equation (8-11) we can find the velocity by knowing the height through which the object has fallen (ignoring drag).

Prelab Questions

What is meant by mechanical energy?

Write an expression for conservation of mechanical energy for an object sliding down a frictionless slope. Compare energies at the top and bottom of the slope.

Write an expression for non-conservation of mechanical energy for an object sliding down a slope with friction. Compare energies at the top and bottom of the slope.

Conservation of Mechanical Energy

Shown in Figure 8-1, is the experimental setup showing the inclined ramp, cart, and photo-gate timer. The height of the incline is given by H = h1 – h2 (h1 data will be provided, h2 = 5 cm and does not change). The distance d = 100 cm and does not change. Here L is the distance that the cart travels down the ramp from its starting position at the top of the ramp to the photo-gate. Place the photo-gate directly over the support h2 as shown in the figure.

Figure 8-1

PROCEDURE

Elevate one end of the ramp to a height h1 as shown in Figure 8-1 (data will be given). Reminder H = h1 – h2

Hold the cart at the top of the ramp and note its position on the ramp ruler scale. Carefully measure the distance L to the photo-gate (you must calculate this distance). This places a photo-gate a distance L down the ramp from the cart’s starting position.

Release the cart from the top of the ramp and determine its instantaneous velocity at the photo-gate (time data will be provided). Be sure to catch the cart at the bottom of the ramp so that it doesn’t bounce back through the gate and effect the reading. (sail length = 0.25 cm)

Repeat step 4 for at least three additional different heights h1 (data provided).

ANALYSIS

Calculate the velocity and kinetic energy for each height H. Tabulate your results in Table 8-1. The mass of the cart = 150 g.

Table 8-1

Trial

Height H (m)

Gate Time (s)

Velocity (m/s)

Kinetic Energy (J)

We will define the potential energy of the cart as zero at the gate position (bottom of hill). Calculate the potential energy at the starting point (top of hill) for each of the four values of H. Tabulate your calculations in Table 8-2.

Table 8-2

Trial

Height H (m)

Potential Energy (J)

Compare your kinetic energy and potential energy values in Table 8-3.

Table 8-3

Trial

Kinetic Energy (J)

Potential Energy (J)

Calculate the velocity using Equation (8-11) and compare with the results in Table 8-1 for each height H. Tabulate your comparisons in Table 8-4.

Table 8-4

Trial

Velocity (m/s) Eq. 8-11

Velocity (m/s) Table 8-1

Dissipative Force and Thermal Energy

Imagine now that the ramp and cart have been used so much by so many physics students that the contact between the cart’s wheels and the ramp surface can no longer be considered to be frictionless. Suppose that the coefficient of kinetic friction, k = 0.10, calculate the thermal energy loss for trial 4.

Calculate the thermal energy due to friction Eth = -fkL, the velocity and the kinetic energy of the cart with the force of friction acting on the cart. Compare the kinetic energies Kfriction and K from Table 8-1. Tabulate your data in Table 8-5.

Table 8-5

Eth (J)

Kfriction (J)

K (J)

Velocityfriction (m/s)

Trial 4

Discussion

Do your values for kinetic and potential energies from Table 8-3 indicate a conservation of mechanical energy? Explain.

With the addition of a frictional force on the ramp/cart, how did this affect the potential and kinetic energies?

CONCLUSION

20F | PHY8103 – Physics 1 Lab 8 – Conservation of Mechanical Energy 6

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