An oversized yo-yo is made from two identical solid disks each of mass M = 2.10

Question

An oversized yo-yo is made from two identical solid disks each of mass M = 2.10 kg and radius R = 9.1 cm. The two disks are joined by a solid cylinder of radius r = 4.00 cm and mass m = 1.00 kg as in the figure below. Take the center of the cylinder as the axis of the system.

What is the moment of inertia of the system? Give a symbolic answer. (Use any variable or symbol stated above as necessary.)

What torque does gravity exert on the system with respect to the given axis

Write an equation for the angular acceleration %u03B1 (alpha) in terms of the translational acceleration a and radius r. (Watch the sign!)

Eliminate %u03B1 from the rotational second law with the expression found in part (f) and find a symbolic expression for the acceleration a in terms of m, M, g, r and R.

What is the numeric value for the system's acceleration?

What is the tension in the string?

How long does it take the system to drop 1.20 m from rest?

Explanation / Answer

(a)The moment of inertia of the system is

I = (MR^2/2) + (mr^2/4)

where M = 2.10 kg,R = 9.1 cm = 9.1 * 10^-2 m,m = 1.00 kg and r = 4.00 cm = 4.00 * 10^-2 m

(b)The torque exerted by gravity is

T = I * g

where g = 9.8 m/s^2

(c)The angular acceleration is

a = r * alpha

or alpha = (a/r)

where a is linear acceleration and r is radius

(d)An expression for a is

a = (M/m) * (R/r) * g

(e)The numerical value is

a = (2.10/1.00) * (9.1/4.00) * 9.8 m/s^2

(f)The tension in the string is

F = (Mm/M + m) * g

(g)let t be the time taken

we know that

S = ut + (1/2)at^2

where u = 0 and S = 1.20 m

or S = (1/2)at^2

or t = (2S/a)^1/2

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