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!
It represents the mechanical energy in joules at the speed gate, at the bottom.
(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
To begin with, let's review the concept of mechanical energy. Mechanical energy is the sum of kinetic energy (Ek) and gravitational potential energy (eg):
Emechanical = Ek + eg
In order to calculate the mechanical energy at the start position and the lowest point of the pendulum's swing, we need to calculate the kinetic energy and gravitational potential energy at those points.
Let's start with the initial position (S). The potential energy at this point is given by the mass of the pendulum bob (m) multiplied by the acceleration due to gravity (g) multiplied by the height (h):
eg = mgh
Given:
Mass of pendulum bob (m) = 240.3 g = 0.2403 kg
Height (h) = 48.0 cm = 0.48 m
Acceleration due to gravity (g) = 9.8 m/s^2
Substituting these values, we have:
eg = (0.2403 kg) * (9.8 m/s^2) * (0.48 m)
eg ≈ 1.134 J
Now, let's calculate the kinetic energy at the lowest point (L) of the pendulum swing. The kinetic energy is given by half of the mass (m) multiplied by the velocity squared (v^2):
Ek = 1/2mv^2
We are given the time interval of photogate light interruption (t), which tells us how much time it took for the pendulum bob to cover a certain distance (d) when it broke the light beam. We can use this information to calculate the velocity (v) of the pendulum bob at the lowest point using the equation v = d/t:
Given:
Time interval (t) = 11.8 ms = 11.8 * 10^-3 s
Diameter of pendulum bob (d) = 3.50 cm = 0.035 m
Substituting these values, we have:
v = d/t = (0.035 m) / (11.8 * 10^-3 s)
v ≈ 2.97 m/s
Now, let's calculate the kinetic energy at the lowest point (L) using the calculated velocity:
Ek = 1/2 * (0.2403 kg) * (2.97 m/s)^2
Ek ≈ 1.134 J
Now, we can calculate the mechanical energy at the start position (S) and the lowest point (L):
Emechanical at S = Ek + eg = 1.134 J + 1.134 J = 2.268 J
Emechanical at L = Ek + eg = 1.134 J + 1.134 J = 2.268 J
Therefore, the mechanical energy at the start position and the lowest point of the pendulum's swing is the same, indicating that mechanical energy is conserved in this system. This confirms the law of conservation of energy.
Now, let's move on to calculating the efficiency of the pendulum as a mechanical machine. Efficiency is defined as the ratio of the useful work output to the total work input. In this case, the useful work output is the mechanical energy at the lowest point (L), and the total work input is the mechanical energy at the start position (S).
Efficiency = (Emechanical at L) / (Emechanical at S) * 100%
Substituting the values, we have:
Efficiency = (2.268 J / 2.268 J) * 100%
Efficiency = 100%
Therefore, the efficiency of the pendulum as a mechanical machine is 100%.
In summary, the pendulum demonstrated the law of conservation of energy by having the same mechanical energy at the start position and the lowest point of its swing. Additionally, the efficiency of the pendulum as a mechanical machine is 100%.