PLEASE SEE ADDITIONAL QUESTION AT BOTTOM.
A pendulum was set up and measurements were made to enable the mechanical energy to be calculated at the start position S and the lowest point of the pendulums swing L.
The mass of the pendulum bob was determined on an electronic scale and its diamter was measured using calipers. The initial height was measured with a meter stick. At the lowest point of its swing, the pendulum bob broke a photogate light beam. The time interval that the light was interrupted was recorded on an electronic timer attached to the photogate.
Use the following data to complete a report.
MAss of pendulum bob = 240.3 g
Diameter of pendulum bob = 3.50 cm
Initial height of pendulum bob = 48.0 cm
Length of pendulum string = 2.14 m
Time interval of photogate light interruption = 11.8 ms
Your report should include the following:
(a) conclusion as to whether or not the pendulum demonstrated the law of conservation of energy
(b) calculations of the efficiency of the pendulum as a mechanical machine
I am really confused how to even start this question. I think that I should figure out the Emechanical = Ek + eg
=1/2(240.3)(0) + 240.3(9.8)(.48)
=1130.37J
but have no idea how to calculate the speed of when it hit. I don't know where to go from here at all, please help!
PHYSICS - Damon, Saturday, January 30, 2010 at 4:10pm
The clue is the diameter of the bob
it broke the beam for 11.8 *10^-3 seconds
it is .035 meters in diameter
so it went
.035 meters in 11.8^10^-3 seconds
which is about 2.97 meters/s
typo, left m out - Damon, Saturday, January 30, 2010 at 4:16pm
(1/2) 240.3 v^2 = mgh = 1130.37
v^2 = 9.407
v = 3.067 ideally but we only measured 2.97 m/s so some energy got lost along the way like air friction and stuff
DAMON: I understand how you got the 1130.37, but don't know how you got v^2. Can you please explain how you got there?...and what are you using for height in the second part of the question...
physics - Damon - bobpursley, Saturday, January 30, 2010 at 5:10pm
1/2 240.3 v^2=1130.37
120.1 v^2=1130.37
v^2= 1130.37/120.1= 9.407
DAMON; thanks for helping so much but I still need some clarification. So the speed when it hit the beam was 2.97 m/s, what does the 3.07 m/s speed represent?
When i subbed in the new speed of 3.07 into the Ek + Eg formula, i got this:
=1/2mv^2 + mgh
=1/2(240.3)(3.07)^2
=1132.4
i am not sure what the 1132.4 actually represents though..please help me more!
physics - Damon - bobpursley, Saturday, January 30, 2010 at 5:36pm
It represents the mechanical energy in joules at the speed gate, at the bottom.
Bob - but if this 1132.4 J is the mechanical energy at the bottom, and the mechanical energy at the top is 1130.37 J, then wouldn't that make is seem like energy was GAINED, rather than lost like DAMON first said?
also,what would percent efficiency equal?
physics - BOB - Damon, Saturday, January 30, 2010 at 6:29pm
The 3.07 is what it should have been if it had all the energy at the bottom that it started with at the top.
It ended up going slower, 2.97, with less energy. Some was lost.
physics - BOB - Damon, Saturday, January 30, 2010 at 6:45pm
(1/2) 240.3 v^2 = mgh = 1130.37
v^2 = 9.407
v = 3.067
this is simply saying that if there is no loss of energy due to friction, the kinetic energy at the bottom (1/2) m v^2 will be the same as the potenital energy at the top m g h
the actual speed measured was less so the actual energy at the bottom is
(1/2) 240.3 (2.97)^2 = 1059.83
so you lost
1130.37 - 1059.83 Joules to friction
and your efficiency is
100 (1130.37 - 1059.83)/1130.37
WOULD PERCENT EFFICIENCY NOT EQUAL:
KINETIC ENERGY GAINED/ENERGY AVAILABLE x 100
so that
1059.83/1130.37 X 100
=94%
because I thought the forumla that you used, Damon, was to calculate percent of energy lost...please clarify! Thanks so much!
SO
To start solving this question, you correctly identified that the mechanical energy of the pendulum can be calculated using the formula E_mechanical = Ek + eg, where Ek is the kinetic energy and eg is the gravitational potential energy. You also correctly calculated the value for eg, which is 1130.37 J.
To calculate the speed of the pendulum when it hits the beam, you can use the information given about the diameter of the bob and the time interval of the light interruption. The diameter of the bob is 3.50 cm, which is equivalent to 0.035 m. The time interval of the light interruption is 11.8 ms, which is equivalent to 11.8 * 10^-3 s.
To calculate the speed, divide the distance traveled (diameter of the bob) by the time interval:
speed = distance / time = 0.035 m / 11.8 * 10^-3 s
Calculating this, you get a speed of approximately 2.97 m/s.
To calculate the value for v^2 (the squared speed), you can use the formula (1/2) mv^2 = mgh, where m is the mass of the pendulum bob, v is the speed, and h is the height.
You already know the value for m (240.3 g) and h (48.0 cm), which you correctly converted to meters (0.48 m). Plugging in these values:
(1/2) * 240.3 * v^2 = 240.3 * 9.8 * 0.48
Canceling out the mass term and solving for v^2:
v^2 = (240.3 * 9.8 * 0.48) / 2
v^2 = 9.407
Taking the square root of v^2, you get approximately v = 3.067 m/s.
However, it is important to note that this value of 3.067 m/s is what the speed should have been if there was no loss of energy due to friction. Since the actual measured speed is 2.97 m/s, it indicates that some energy was lost, likely due to air friction and other factors.
Therefore, the actual mechanical energy at the bottom is given by:
E_actual = (1/2) mv^2 = (1/2) * 240.3 * (2.97)^2
E_actual = 1059.83 J
This means that the loss of energy can be calculated as:
Energy lost = E_mechanical - E_actual = 1130.37 J - 1059.83 J = 70.54 J
To calculate the percent efficiency of the pendulum as a mechanical machine, you can use the formula:
Percent efficiency = (Energy lost / E_mechanical) * 100
Using the values we obtained:
Percent efficiency = (70.54 J / 1130.37 J) * 100 ≈ 6.24%
So the percent efficiency of the pendulum as a mechanical machine is approximately 6.24%.