A yo-yo is made from two uniform disks, each with mass m and radius R , connecte

Question

A yo-yo is made from two uniform disks, each with mass m and radius R, connected by a light axle of radius b. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds.

Part A:

Find the acceleration of the yo-yo (Express your answer in g, b, R, m)

Part B: Find the angular acceleration of the yo-yo (Express your answer in g, b, R, m)

Part C: Find the tension in the string (Express your answer in g, b, R, m)

Explanation / Answer

m be the mass of each disk,
R be the radius of each disk,
b be the radius of the light axle,
g be the acceleration due to gravity,
T be the tension in the string,
I be the moment of inertia of the yo-yo about its centre,
alpha be the angular acceleration.

The equation of motion for the centre of mass is:
2mg - T = 2ma ...(1)

The equation for the rotational motion is:
I alpha = bT ...(2)

In addition:
a = b alpha ...(3)
I = mR^2 ...(4)

Eliminating T from (1) and (2):
I alpha = 2mb(g - a) ...(5)

Eliminating alpha from (3) and (5):
Ia = 2mb^2 (g - a)
(I + 2mb^2)a = 2mb^2 g
a = 2mb^2 g / (I + 2mb^2).

1.
Substituting for I from (4):
a = 2b^2 g / (R^2 + 2b^2) ...(6)

2.
Eliminating a from (3) and (6):
alpha = 2bg / (R^2 + 2b^2)

3.
From (1) and (6):
T = 2m(g - a)
= 2mg[ 1 - 2b^2 / (R^2 + 2b^2) ]
= 2mgR^2 / (R^2 + 2b^2).

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.