Two circular wheels, the first of radius R and mass M and the second of radius r

Question

Two circular wheels, the first of radius R and mass M and the second of radius r and mass m, are connected by a light, inextensible drive belt, Both wheels start from rest at t = 0 and an external torque is supplied to the first wheel so that it is given a constant angular acceleration of a1 (in the counter-clockwise, or positive, direction). The drive belt will not slip on either wheel, and first wheel will drive the second wheel. The wheels can be treated as disks of uniform density.

(a) Write down an expression for the angular displacement of the first wheel from its starting position after t seconds.

(b) What is the kinetic energy of the first wheel at t secodns?

(c) Find an expression for the angular acceleration of the second wheel, a2

(d) Find an expression for Tu-Tl the difference between the tensions in the upper (Tu) and lower (Tl) parts of the drive belt.

(e) What is the external torque on the first wheel that must be supplied to cause both wheels to rotate as described? Give your answer in terms of M, R, m, and a1

Could anyone help me with this problem?

Two circular wheels, the first of radius R and mass M and the second of radius r and mass m, are connected by a light, inextensible drive belt, Both wheels start from rest at t = 0 and an external torque is supplied to the first wheel so that it is given a constant angular acceleration of a1 (in the counter-clockwise, or positive, direction). The drive belt will not slip on either wheel, and first wheel will drive the second wheel. The wheels can be treated as disks of uniform density. (a) Write down an expression for the angular displacement of the first wheel from its starting position after t seconds. (b) What is the kinetic energy of the first wheel at t secodns? (c) Find an expression for the angular acceleration of the second wheel, a2 (d) Find an expression for Tu-Tl the difference between the tensions in the upper (Tu) and lower (Tl) parts of the drive belt. (e) What is the external torque on the first wheel that must be supplied to cause both wheels to rotate as described? Give your answer in terms of M, R, m, and a1 Could anyone help me with this problem?

Explanation / Answer

a)

Here , for the first wheel ,

at t = t s

w = a1*t

Therefore

KE = 0.5*I*w^2

KE1 = 0.5 * 0.5*M*R^2 * (a1*t)^2

KE1 = 0.25MR^2*(a1*t)^2

b)

here ,

a1 * R = a2 * r

a2 = a1*R/r

Here , as the tension drives the smaller wheel

(Tu - Tl) * r = 0.5 * mr^2 * a1*R/r

(Tu - Tl) = 0.5m*a1*R

c)

Using second law of motion

T = (Il*a1 + Is*a2)

T = 0.5MR^2 *a1 + 0.5mr^2 * a1*R/r

T = 0.5MR^2*a1 +0.5mRr*a1

the external torque on the first wheel is 0.5MR^2*a1 +0.5mRr*a1

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