In class, we've been working on moment of inertia and recently did a lab regarding its usage. I don't feel too confident with the material yet and was hoping someone could look over my work and offer advice on what to do.

We used solid spheres in the lab (2/5mr^2 used) and measured the height from the object was released, the length of the ramp, radius from the axis of rotation, mass, and time for the object to reach the bottom.

Object 1: h=0.085 m, l=0.7 m, r=0.05 m, m=0.81 kg, t=1.3 s
Object 2: h=0.085 m, l=0.7 m, r=0.025 m, m=0.11 kg, t=1.2 s

Calculations:
1.) Speed of the object at the bottom of the ramp if all it did was slide down (disregard friction).
mgh=1/2mv^2
gh=1/2v^2
v=(2*9.8*0.085)^1/2=1.29 m/s

2.) Speed of the object if it was rotating down the ramp (no slipping).
mgh=1/2mv^2 + 1/2Iω^2
mgh=1/2mv^1 + 1/2(2/5mr^2)(v^2/r^2)
mgh=1/2mv^2 + 1/5mv^2
gh=7/10v^2
v=(10/7*9.8*0.085)^1/2=1.09 m/s

3.) Calculate the angular speed of the object at the bottom of the ramp.
ω=v/r
ω1=1.09/0.05=21.8 rad/s
ω2=1.09/0.025=43.6 rad/s

4.) Assuming no slipping, calculate the number of revolutions it takes for the object to reach the bottom.
Ө=1/2(ωf+ωi)t
Ө1=1/2(21.8+0)(1.3)=14.17 rad=2.26 rev
Ө2=1/2(43.6+0)(1.2)=26.16 rad=4.16 rev

5.) Using #4, calculate Ө (rad).
Ө1=14.17 rad
Ө2= 26.16 rad

6.) Calculate α (rad/s^2).
21.8=0 + 1.3α
α1=16.77 rad/s^2
43.6=0 + 1.2α
α2=36.33 rad/s^2

7.) Calculate t (s).
14.17=0 + 1/2*16.77*t^2
t1=1.3 s
26.16=0 + 1/2*36.33*t^2
t2=1.2 s

However, I know that the time calculated should be different than the time measured and I'm not sure what to do differently...

UPDATE: On 4, I tried using Δx=rΔӨ. It cleared up a lot of data issues.

Great job on your calculations! It seems like you have a good understanding of the formulas and concepts related to moment of inertia. However, the difference between the calculated time and the measured time could be due to a few factors:

1. Friction: In your calculations, you assumed no friction. In reality, there may have been some friction present, which could affect the object's speed and time. Try accounting for friction in your calculations to see if it brings the calculated time closer to the measured time.

2. Experimental Errors: It's common for experimental measurements to have some degree of error. Make sure you have taken accurate measurements of the height, length of the ramp, radius, mass, and time. Perhaps double-check your measurements and make sure they are as precise as possible.

3. External Factors: Consider other external factors at play during the experiment, such as air resistance, which could have affected the object's motion. Although you assumed these factors were not present in your calculations, they may have influenced the object's behavior in reality.

To improve your calculations and compare them more accurately to the measured time, I would recommend addressing these factors:

1. Account for Friction: Research or ask your instructor about the effects of friction on the object's motion down the ramp. Modify your calculations to include the frictional forces, which will help you obtain a more accurate result.

2. Minimize Experimental Errors: Make sure to measure all the variables with high accuracy. Consider using more precise measuring tools or techniques to improve the accuracy of your measurements.

3. Evaluate External Factors: Research if any external factors, such as air resistance, could affect the motion down the ramp. If necessary, incorporate these factors into your calculations to obtain a more realistic result.

By considering these factors and fine-tuning your calculations, you should be able to achieve a closer match between the calculated time and the measured time. Remember, experimental data can often deviate from theoretical calculations due to various factors, so don't be discouraged if they are not a perfect match.