A cylinder with radius r and moment of inertia / about its central axis is mount

Question

A cylinder with radius r and moment of inertia / about its central axis is mounted ona frictionless bearing and wrapped with a massless string. The string leads over a massless, frictionless pulley to a mass m hanging in the air. The cylinder starts from rest and begins to rotate as the mass accelerates downward. 1. a) (2 pts.) Find the string's tension in terms of the hanging mass m and its acceleration a b) (3 pts.) Find the rotational inertia / by measuring the hanging mass m, its acceleration a, and radius r. Express / in terms of m, a, and r. c) (3 pts.) Using this experimental setup, you measure m, a, and r. You also record the uncertainty in each of these measurements. The results of these measurements are m = 0.5 ± 0.0001 kg, r = 12.75± 0.005 cm, and a-12 ± 0.1 m/s2, when you combine these measurements to find I, which measurement uncertainty dominates the uncertainty in 1? You must give a quantitative argument for your answer. d) (2 quantities you actually measure and their associated uncertainties. Hint: If there is one uncertainty that is much larger than all the others, it is only one you need to consider. pts.) Find an expression for the uncertainty in the moment of inertia, , in terms of the

Explanation / Answer

1. (a) on hanging mass, Fnet = m a

m g - T = m a

T = m ( g - a)

(b) torque = I alpha

r T = I (a / r)

I = T r^2 / a

I = m r^2 (g - a) / a

(c) I = 0.5 x 0.1275^2 (9.80 - 1.2) / 1.2

I = 0.058 kg m^2

&I = I sqrt[ (0.0001/0.5)^2 + 2(0.005/12.75)^2 + 2(0.1/1.2)^2]

& I = 0.007

(D) &I = (m r^2 (g - a) / a ) sqrt[ (&m/m)^2 + 2(&a/a)^2 + 2(&r/r)^2]

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