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Conservation of Mechanical Energy – Procedure

Objective

In this experiment, you will roll a ball down a ramp situated above a table as in Figure 1.
A meterstick and vertical rod are resting on a tabletop.  A metal track is attached to the rod by an adjustable clamp such that the lower end is positioned above the table. A ball is located at upper end of the track.  The lower end of the track bends into a flat section oriented parallel to the surface of the table.
Figure 1
You will measure the horizontal and vertical distances associated with the motion in order to determine the speed of the ball at the time it leaves the ramp. You will use two different methods for this experiment. These methods are discussed in more detail below.

Equipment

  • Metal track and stand
  • Marble
  • Meterstick
  • Carbon paper
  • White paper
  • Tape

Background

The objective of this experiment is to determine if 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 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 Figure 2, you can show that the final speed
v2
at the end of the ramp is
( 1 )
v2 =
10gh1
7
.
A thin horizontal line spans the bottom of the figure.  The lower end of a diagonal line is located a distance h_2 above the thin horizontal line toward the middle of the figure.  A solid circle is located at the upper end of the diagonal near a point labeled 1.  A horizontal line is connected to the lower end of the diagonal at a point labeled 2 and extends a short distance to a point labeled 3.  The vertical distance between points 1 and 2 is labeled h_1.  A vertical dashed line extends from point 3 down to the thin horizontal line.  A point labeled 4 is located on the thin horizontal line a distance d from the dashed vertical line in the direction away from the other three points.
Figure 2
Equation 1 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 conservation of mechanical energy.

Experimental Design

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

Finding the speed using kinematics

The ramp has a short horizontal section from point Point two to point Point three. 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 Point three. 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
( 2 )
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
( 3 )
d = v2(ΔT).
Using equation 2 and equation 3, we can show that
( 4 )
v2kinematics = d
g
2h2
.

Procedure

Please print the worksheet for this lab. You will need this sheet to record your data.

PDF file

Kinematics

The first method will apply the principles of uniformly accelerated motion to treat the ball as a projectile. Measuring d and h2 as illustrated in Figure 2 will be enough to calculate the velocity of the ball at the time it leaves the ramp. As a reminder, the uniformly accelerated motion equations are reproduced below. For more details on this method, refer to the Concepts document.
( 5 )
vx = vx,0 + axt        vy = vy,0 + ayt
x = x0 + vx,0t +
1
2
axt2
        y = y0 + vy,0t +
1
2
ayt2
vx2 = vx,02 + 2ax(xx0)        vy2 = vy,02 + 2ay(yy0)
1
Arrange the track and stand as illustrated in Figures 1, 2, and 3. Make sure the bottom part of the track is flat and that the ball won't roll off the sides on its way down.
2
Hold the ball steady at a point on the track and record the vertical distance from the table to the ball. (Hint: Should you measure to the top, middle, or bottom of the ball?) As illustrated in Figure 3, the height, h1, will be the difference between this distance and h2, the distance from the table to the flat part of the track.
3
Release the ball and record where it lands. One method of doing this is to use the supplied carbon paper. If you use carbon paper, put a blank piece of paper down where the ball will land, then the carbon paper black-side-down on it. The ball's impact will make a mark on the paper. If you use the paper, make sure to hold or tape it still so it doesn't slide when the ball impacts.
A vertical line extends upward from one end of a horizontal rectangle labeled Table.  A diagonal line intersects the vertical line at its midpoint such that the lower end is located a distance h_2 above the Table.  The intersection point is surrounded by a small rectangle labeled Adjustable clamp.  A solid circle is located at upper end of the diagonal at a point labeled 1.  A horizontal line labeled Ramp is connected to the lower end of the diagonal at a point labeled 2 and extends a short distance to a point labeled 3.  The vertical distance between points 1 and 2 is labeled h_1.  Another solid circle is located at the opposite end of the Table at a point labeled 4.  The horizontal distance between point 4 and point 3 is labeled d.
Figure 3
The horizontal distance from the edge of the ramp to the impact point will be d as illustrated in Figure 3. The vertical distance from the ramp to the table surface will be h2.
4
In order to improve the accuracy of the data, you will perform this measurement two more times using the same starting height (that is, the same h1), finding the average value for d over the trials. Then for further accuracy, you will repeat the experiment with different starting locations, performing three trials for each different h1.
5
Finally, use the kinematics equations and the collected data to determine the velocity of the ball at the instant it left the ramp.

Energy

Next, you will calculate the speed of the ball as it leaves the ramp again, this time using conservation of energy methods. During the ball's roll down the ramp, we will assume that it is rolling without slipping. This would mean that friction, while present, is not doing any work. If we also ignore the very small contribution due to air resistance, the conclusion is that the ball has no nonconservative force doing work on it and thus its mechanical energy is conserved. As a reminder, here are the expressions for various types of mechanical energy.
( 6 )
Ktrans =
1
2
mv2
( 7 )
Krot =
1
2
Iω2
( 8 )
Ug = mgy
Note that for a solid sphere,
I =
2
5
MR2.
1
Using the data you collected in the Kinematics section, calculate the velocity at the bottom of the ramp for each starting height. Use the starting position (at the top of h1) and the end of the ramp as your initial and final points. Should these calculations yield the same results as the kinematics calculations?

Conclusion

1
What are some of the sources of uncertainty in this lab that could have contributed to a discrepancy in the two data sets or to one or both of the calculations being too high or low? Specify which calculation method each source of uncertainty would contribute to, and whether it would tend to make the calculation too low or too high.
2
Considering all the sources of uncertainty you identified above, which method of calculation do you feel is likely to give a more accurate result? Consider how many sources of uncertainty each method might have and the magnitude of those uncertainties in your conclusion.