show all your work please!! 3. 10 points: The radius of the roll of paper as sho

Question

show all your work please!!

3. 10 points: The radius of the roll of paper as shown in the figure is r and its moment of inertia is I. A force F is exerted on the end of the roll for time trull, but the paper does not tear so it begins to unroll. A constant friction torque of Tfrie is always present; after tpull it gradually brings the roll to a stop. Assuming that the paper's thickness is negligible, calculate 2014 Poarsor Education, Inc. Figure 3: Diagram for Problem 3 (a) the length of paper that unrolls during the time that the force is applied (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving. (c) Now solve a. and b. in the case that r = 6cm, 1-3 × 10-3kg. m2, F = 3.5N, tpull = 1.5s and Tjric = 0.11mM

Explanation / Answer

3. radius = r

moment of inertia = I

Force = T

time = tpull

frictional torque = Tf

a. let length of paper that unrolls in the time tpull be L

angular acceleration for the duration = alpha

Tr - Tf = I*alpha

and

L/r = 0.5*alpha*tpull^2 = 0.5*(Tr - Tf)*tpull^2/I

L = (Tr^2 - Tf*r)*tpull^2/I

final angular velocity = w

w = alpha*tpull = (Tr - Tf)*tpull/I

b. length of paper that unrolls after the force is removed = l

angula acceleration = beta

2*beta*l/r = w^2

Tf = I*beta

hence

l = r(Tr - Tf)^2*tpull^2/2*Tf*I

c. r = 6 cm = 0.06 m

I = 3*10^-3 kg m^2

T = 3.5 N

tpull = 1.5 s

Tf = 0.11*10^-3 N

hence

L = 9.44505 m

l = 9011.007 m

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