(a) Show that the moment of inertia of a uniform disk of radius r and mass m, [4

Question

(a) Show that the moment of inertia of a uniform disk of radius r and mass m, [4 marks] Q4 through the central axis, is I = mr2/2. (b) A yo-yo is made from two uniform disks, each with mass m and radius R connected by a light cylidrical 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. Show that the angular acceleration, a of the yo-yo is 2g and the tension in the string is 2mg 2(b/R)2 + 1 8 marks 4 marks [4 marks For a string length of 1.0 m, and with R 10 cm and b 1.0 cm, calculate the vertical speed of the yo-yo just as the string finishes unwinding. (c) Explain, in energy terms, how a correctly deployed yo-yo is able to return back up to the hand.

Explanation / Answer

Q4.

a. consider a uniform disc of mass m and radius r

consider a ring of thickness dx and radius x < r on this disc

moment of inertia of this ring, dI = dm*x^2

where dm = m*2*pi*x*dx/pi*r^2 = 2m*xdx/r^2

so dI = 2mx^3dx/r^2

integrating from x = 0 to x = r

I = 2mr^4/r^2*4 = mr^2/2

b. a YO Yo , made up of two uniform discs , each of mass m and radius R

radius of axle = b

when the YO Yo is left

let tnesion in the string be T

then from force balance

2mg - T = 2ma

but T*b = I*alpha

where I = mR^2

and alpha = a/b

hence

2mg - mR^2*alpha/b = 2m*alpha*b

alpha = 2g/(R^2/b + 2b)

tension in the string

T = 2mg - 2ma = 2m(g - a) = 2m(g - 2gb/(R^2/b + 2b))

T = 2m(g - a) = 2mg(1 - 2b^2/(R^2 + 2b^2))

T = 2mg(R^2 + 2b^2 - 2b^2)/(R^2 + 2b^2)

T = 2mg/(1 + 2b^2/R^2)

l = 1 m

R = 0.1 m

b = 0.0 1m

alpha = 19.235 rad/s

a = alpha*b = 0.192 m/s

so final velocity, when the string finishes unwinding = v

2*a*l = v^2

2*0.192*1 = v^2

v = 0.619 m/s

c. as there is no energy wasted anywhere, so total PE of the YO YO in the begining is efficiently converted forst to its KE and then back to its PE when the YOYO comes back to its initial position

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