Download Conservation of Mechanical Energy

PC1221 Fundamentals of Physics I
Conservation of Mechanical Energy
1
Purpose
• Determine the change in the gravitational potential energy ∆U as a cart moves down
an inclined dynamic track.
• Determinate the change in the kinetic energy ∆Ek as a cart moves down an inclined
dynamic track.
• Evaluate the extend to which the total mechanical energy (sum of kinetic energy plus
gravitational potential energy) is conserved.
2
Equipment
• Dynamics cart and track
• Photogates
• Adjustable end stop
• Smart Timer
• Picket fence
• Adjustable ladder
• Balance and meter rule
3
Theory
The mechanical energy E of a system is the sum of its potential energy U and the kinetic
energy Ek of the objects within it
E =K +U
(1)
The kinetic energy of an object is given by
Ek =
Level 1 Laboratory
1
mv 2
2
Page 1 of 4
(2)
Department of Physics
National University of Singapore
Conservation of Mechanical Energy
Page 2 of 4
where m is the mass of the object and v is the speed of the object. The gravitational potential
energy of an object, which is not too far away from the ground, is given by
U = mgh
(3)
where g is the acceleration due to gravity and h is the height of the object above the position
where the potential energy is defined to be zero.
When a conservative force does work W on an object within the system, one of these
energies increases exactly as much as the other decreases. In an isolated system where only
conservative forces cause energy changes, the kinetic energy and potential energy can change,
but their sum, the mechanical energy E of the system, cannot change. This result is called
the principle of conservation of mechanical energy. We can write this principle in a compact
form
∆E = ∆Ek + ∆U = 0
(4)
The principle of conservation of mechanical energy allows us to solve problems that would
be quite difficult to solve using only Newton’s laws. When the mechanical energy of a system
is conserved, we can simply relate the sum of the kinetic energy and potential energy at one
instant to that at another instant without considering the intermediate motion and without
finding the work done by the forces involved.
In this experiment, we will examine the transformation of energy that occurs as a dynamic
cart slides down an inclined track. Since there are no objects to interfere with the motion and
there is minimal friction between the track and cart, the loss in gravitational potential energy
as the cart slides down the track should be very nearly equal to the gain in kinetic energy.
Stated mathematically:
∆Ek = ∆U = mg∆h
(5)
where ∆h is the change in the vertical position of the cart. Here, ∆Ek is the change in kinetic
energy of the cart
1
1
(6)
∆Ek = mv22 − mv12
2
2
where v1 and v2 are the speeds of the cart at higher (initial) and lower (final) positions
respectively.
4
Experimental Procedure
P1. Setup the photogates and Smart Timer as shown in the Figure 1. Position the two
photogates just far enough apart.
P2. Put a picket fence into the slot at the top of the cart. Adjust the top of the height of
the photogates so that the 1-cm flag will block the photogate infrared beams. Connect
the photogates to the Smart Timer.
Level 1 Laboratory
Department of Physics
National University of Singapore
Conservation of Mechanical Energy
Page 3 of 4
Figure 1: Experiment setup for conservation of mechanical energy
P3. Setup the Smart Timer to measure Speed: collision (cm/s). Press Start/Stop to
activate the Smart Timer.
Note: If the flag does not go through the photogate beams twice, the Smart Timer
will not complete the timing cycle and display velocities automatically. You will need
to press Start/Stop to stop timing manually. The completed timing measurements
will be displayed and the uncompleted measurements will be registered as 0. If the cart
passes through the photogates just once, take the non-zero reading as the velocity of the
cart when the cart passes through the corresponding photogate. Press 1 or 2 to scroll
back and forth between the registered velocities from respective photogates.
P4. Measure the distance the cart moves on the dynamic track between the photogates.
Record this distance as d in Data Table 1.
Caution: The distance d has to be kept constant throughout the experiment.
P5. Using a laboratory balance, measure the mass of the cart (picket fence inclusively).
Record this mass as m in Data Table 1.
P6. Place the adjustable ladder under one end of the dynamic track.
P7. Measure the distance between two ends of the inclined dynamic track. Record this
distance as D in Data Table 2.
P8. Measure the height of the inclined dynamic track above the surface of the table. Record
this height as h in Data Table 2.
Level 1 Laboratory
Department of Physics
National University of Singapore
Conservation of Mechanical Energy
Page 4 of 4
P9. Hold the cart steady near the top of the inclined dynamic track, then release it so it
moves freely through the photogates. Record the velocities v1 and v2 , the velocities
during which the cart passes the photogate 1 and 2 respectively, in Data Table 2.
P10. Repeat the measurements of velocities v1 and v2 THREE times and record your data
in Data Table 2. You need not release the cart from the same point on the dynamic
track for each trial, but it must be moving freely and smoothly as it passes through the
photogates.
P11. Adjust the adjustable ladder to have FIVE readings of the height. For each trial, repeat
the steps P8–P10.
5
Data Analysis
D1. Calculate the angle of incline for the dynamic track θ for each trial in Data Table 2.
D2. Calculate the change in vertical position of the cart when passing between the photogates
∆h for each trial in Data Table 2.
D3. Calculate the loss in gravitational potential energy of the cart mg∆h when passing
between the photogates for each trial in Data Table 2.
D4. Calculate the gain in kinetic energy of the cart ∆Ek when passing between the photogates for each trial in Data Table 2.
D5. Calculate the percentage difference between the loss in gravitational potential energy
mg∆h and gain in kinetic energy ∆Ek of the cart when passing between the photogates
for each trial in Data Table 2.
Hint: The percentage difference between two measurements is a measure of the precision
and is defined as
Percentage difference =
|E2 − E1 |
× 100%
|E1 + E2 |/2
where E1 and E2 are two different experimental values.
Level 1 Laboratory
Department of Physics
National University of Singapore