homework
1) In a paragraph, describe how to derive an equation for gravitational acceleration in terms of the masses hanging on the string, distance the masses travel (one mass will ascend and the other will descend the same distance) and time it takes the massed to travel that distance. If you want to use the Canvas “Insert Math Equation” feel free to do so.
In this case, we have two masses say m1 and m2 attached with a string on a pulley. The mass of the pulley can be denoted as M. Newton approach is used to find the motion. Assume in this case, m1 > m2, the forces can be, m1g and m2g. Tension associated will be T1 and T2 which are acting on the opposite directions in both cases. The first equation will be m1g – T1 = m1a and the second will have a higher tension compared its force hence, T2 – m2g = m2a. We should also consider the motion of the rotating pulley which is given as = (T1 – T2) * R where R is the radius of the pulley and the rotation is therefore = (T1 – T2)R. By definition = the moment of inertia and its given as I = ½ MR2where R is the radius of the pulley. We hence add our first two equations to have, m1g – T1 + T2 – m2g = m1a + m2a = (m1-m2)g – (T1 – T2) = (m1+m2)a
Remember = (T1 – T2)R and = I, therefore I = (T1 – T2)R which implies that ½ mpR2α = (T1 – T2)R, divide through by R to get, ½ MRα = (T1 – T2). No slipping condition is given as V = Rw where derivative will give time as a = Rα. Make α subject of the formula and substitute it ½ MRα = (T1 – T2) to give (T1 – T2) = ½ MR * a/R which gives (T1 – T2) = ½ Ma. Therefore we have (m1 – m2)g – (T1-T2) = (m1+m2)a which gives (m1 – m2)g – ½ Ma = (m1+m2)a. This can be simplified to give;
(m1 – m2)g = (m1+m2+½ M)a, making a subject of the formula we have;
a = which is our acceleration equation.
2) Show one example of a numerical calculation of gravitational acceleration using the equation you derived in 1) above and one set of data that you collected.
The equation formed is a =
Suppose we have pulley mass M = 32kg, m1 = 25kg, m2 = 15kg, radius = 6m
Acceleration will be given as a = = 1.75 m/s2
3) Show all calculated values for gravitational acceleration from all experimental runs. There should be data from 10 different configurations. There is no need to include the computational steps for each value of gravitational acceleration as that was demonstrated in 2) above, just list the final calculated value.
|
m1 |
m2 |
Time |
Distance |
|
1.1 |
2.1 |
0.063 |
25 cm |
|
2.2 |
3.1 |
0.038 |
30 cm |
|
3.2 |
4.3 |
01.81 |
45 cm |
|
4.3 |
5.4 |
02.39 |
50 cm |
|
5.4 |
6.4 |
01.16 |
55 cm |
|
6.4 |
7.4 |
00.86 |
60 cm |
|
6.4 |
8.5 |
01.25 |
65 cm |
|
8.5 |
9.6 |
01.31 |
70cm |
|
9.6 |
10.7 |
01.88 |
75 cm |
|
10.7 |
11.7 |
02.83 |
80 cm |
4) Do your experimentally determined values of gravitational acceleration match 9.8 m/s^2? If so what experimental factors made the results work well? If not, what experimental factors caused the results to be off?
No. Different masses, pulley mass and radius and frictions are some of the factors that caused the results to deviate from 9.8 m/s2