Solve for T: Solving "T" in equation - Period Calculation

[riseofphoenix]

What is the period "T" in this equation? Or how do I solve for it?

1)

This is what I did.

a_c (first column on excel)

a_c = v²/r
a_c = (2πr/T)²/r

m_hg (second column on excel)
0.07155(9.81) = 0.7019055
0.10422(9.81) = 1.0223982
0.11394(9.81) = 1.1177514
0.13392(9.81) = 1.3137552
0.15795(9.81) = 1.5494895

 

[SHISHKABOB]

okay so the period of the Uniform Circular Motion of the bob is the time that it takes the bob to go around the circle *once*. A complete rotation takes an amount of time that is equal to the length of time of the period.

If the period is 5 seconds, then the bob takes 5 seconds to go around the circle once. In other words, T = 2πr/v like you have it. This is exactly the same thing as t = x/v for regular motion. Since Distance equals Rate times Time, we have Circumference (the distance that the bob travels in the circle that it's making) equals Rate times Period. So T = 2πr/v, and then you can substitute that into the centripetal acceleration like you've done.

So then, if the bob has traveled around the circle 50 times, what would the distance traveled be? If we know that the time taken to travel a distance equal to the circumference is equal to T, then what would be the time taken to travel 50 times that distance (50 times around the circle)?

 

[riseofphoenix]

Distance = r x t (time)
and for this particular problem
Circumference = = r x T (period)

 

[SHISHKABOB]

Well so the time that you are given is the time taken for the bob to go around 50 times. The period is the time taken to go around *once*. It takes the bob 50 times as long to go around 50 times as it does to go around once.

 

[riseofphoenix]

Well so the time that you are given is the time taken for the bob to go around 50 times. The period is the time taken to go around *once*. It takes the bob 50 times as long to go around 50 times as it does to go around once.

99.77 s -------------- 50 times
T -------------------- 1 time

So,
50 T = 99.77
T = 99.77/50
T = t/50 (you do this for the other "t" values)

a_c (first column in excel)

T = 99.77/50
T = 1.9954 for v₁

v₁ = 2π(0.1825)/1.9954 = 0.5746
a_c1 = (0.5746)²/(0.1825) = 1.809
__________________________________________________

T = 90.61/50
T = 1.8122 for v₂

v₂ = 2π(0.2430)/1.8122 = 0.8379
a_c2 = (0.8379)²/(0.2430) = 2.889

__________________________________________________

T = 92.31/50
T = 1.8462 for v₃

v₂ = 2π(0.2610)/1.8462 = 0.888
a_c3 = (0.888)²/(0.2610) = 3.023

__________________________________________________

T = 88.52/50
T = 1.7704 for v₄

v₄ = 2π(0.2980)/1.7704 = 1.0576
a_c4 = (1.0576)²/(0.2980) = 3.753

__________________________________________________

T = 87.38/50
T = 1.7476 for the v₅

v₅ = 2π(0.3425)/1.7476 = 1.231
a_c5 = (1.231)²/(0.3425) = 4.427

__________________________________________________

 

[SHISHKABOB]

yes that is the correct calculation

 

[riseofphoenix]

yes that is the correct calculation

So this is what the graph looks like.

Answer: the mass of the bob (the slope of the line) is 0.3224 kg