Two discs are fastened together, so they act as one object with a common axis of
ID: 1616147 • Letter: T
Question
Two discs are fastened together, so they act as one object with a common axis of rotation. R_1 = 3 m, and R_2 = 4.5 m. Each disc is 10 cm thick and is constructed of a material with a density of 20 kg/m^3. Two masses hang from thin cables, as shown. a) Calculate the moment of inertia of the composite disc. b) If M_1 = 66 kg, what is M_2 such that no angular acceleration results in the system? c) If 33 kg is added to M_1, and M_2 is the same as part b) calculate the angular acceleration of the discs, the linear accelerations of both masses and the tensions in both cables.Explanation / Answer
(A)
V = pi r^2 d
m = rho V
I = m r^2 /2 = (rho pi r^2 d) r^2 / 2
I = rho pi r^4 d / 2
Inet = I1 + I2
= (rho pi d / 2) [ R1^4 + R2^4]
= (20 x pi x 0.10 / 2) [ 3^4 + 4.5^4]
= 1542.72 kg m^2
(b) no angular acc means net torque zero.
torque = M1 R1 g - M2 R2 g = 0
66 x 3 = M2 x 4.5
M2 = 44 kg
(c) now net torqe will be due to extra added mass,
torque = 33 x 3 x 9.8 = 970.2 N m
torque = I alpha
alpha = 970.2 / 1542.72 = 0.63 rad/s^2 ..........Ans
a1 = alpha r1 = 1.89 m/s^2 ......Ans
a2 = alpha R2 = 2.83 m/s^2 .....Ans
on M1.
M1g - T1 = M1 a
T1 = (33 + 66) (9.8 - 1.89)
T1 = 783 N
on M2:
T2 - M2g = M2 a
T2 = 44 ( 9.8 + 2.83) = 555.72 N ........Ans
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