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...
1/2 240.3 v^2=1130.37
120.1 v^2=1130.37
v^2= 1130.37/120.1= 9.407
(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 calculate the speed of the pendulum bob at the lowest point of its swing, you can use the conservation of mechanical energy equation.
The equation is:
E_mechanical = E_kinetic + E_gravitational potential
Where:
E_mechanical is the total mechanical energy
E_kinetic is the kinetic energy
E_gravitational potential is the gravitational potential energy
In this case, the gravitational potential energy can be calculated as mgh, where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the height.
Given that:
m = 240.3 g (convert to kg by dividing by 1000: 0.2403 kg)
g = 9.8 m/s^2
h = 48.0 cm (convert to meters by dividing by 100: 0.48 m)
E_gravitational potential = 0.2403 kg * 9.8 m/s^2 * 0.48 m = 1.128 J
The total mechanical energy E_mechanical is equal to the sum of the kinetic energy E_kinetic and the gravitational potential energy E_gravitational potential.
E_mechanical = E_kinetic + E_gravitational potential
Since the pendulum bob is at the lowest point of its swing, the gravitational potential energy is at its minimum and the kinetic energy is at its maximum.
So the equation can be written as:
E_mechanical = E_kinetic (since E_gravitational potential = 0)
E_mechanical = (1/2) m v^2 (where v is the velocity/speed of the pendulum bob)
Therefore:
(1/2) m v^2 = 1.128 J
Plugging in the value for m:
(1/2) * 0.2403 kg * v^2 = 1.128 J
Now, to find v^2, we can rearrange the equation:
v^2 = (2 * 1.128 J) / 0.2403 kg
v^2 = 9.392 m^2/s^2
Finally, taking the square root of both sides gives us the speed of the pendulum bob:
v = √(9.392 m^2/s^2) ≈ 3.067 m/s
This shows that the initial calculated speed of the pendulum bob is approximately 3.067 m/s. However, in this case, the measured value is given as 2.97 m/s. This difference can be attributed to factors like air friction and other losses of mechanical energy during the swing.
For the second part of the question, to calculate the efficiency of the pendulum as a mechanical machine, you can use the formula:
Efficiency = (useful energy output / total energy input) * 100
In this case, the useful energy output is the kinetic energy at the lowest point of the swing, which is given by (1/2) m v^2.
The total energy input is the initial mechanical energy at the start position, which is given by (1/2) m v_initial^2.
Therefore, the efficiency of the pendulum can be calculated as:
Efficiency = [(1/2) m v^2 / (1/2) m v_initial^2] * 100
Simplifying the equation:
Efficiency = [(v^2) / (v_initial^2)] * 100
Substituting the values:
Efficiency = [(2.97 m/s) / (3.067 m/s)] * 100
Efficiency ≈ 96.82%
This means that the efficiency of the pendulum as a mechanical machine is approximately 96.82%.