Two objects, of masses m1 = 485.0 g and m2 = 545.7 g, are connected by a string

Question

Two objects, of masses m1 = 485.0 g and m2 = 545.7 g, are connected by a string of negligible mass that passes over a pulley with frictionless bearings. The pulley is a uniform 45.5-g disk with a radius of 3.64 cm. The string does not slip on the pulley. (a) Find the accelerations of the objects. (b) What is the tension in the string between the 485.0-g block and the pulley? (Round your answer to four decimal places.) What is the tension in the string between the 545.7-g block and the pulley? (Round your answer to four decimal places.) By how much do these tensions differ? (c) What would your answers be if you neglected the mass of the pulley? Acceleration? Tension?

Explanation / Answer

FBD of m2 yields: m2*g - T2 = m2*a

FBD of m1 yields: T1 - m1*g = m1*a

Adding both equations: (T1 - T2) + (m2 - m1)g = (m1 + m2)a

Or, (T2 - T1) = (m2-m1)g - (m1+m2)a............(1)

Torque on pulley = (T2 - T1)*r = I where I is inertia of pulley and is angular acceletation.

For disk, I = 1/2*mr^2

Also, = a/r

So, (T2 - T1)*r = 1/2*mr^2*(a/r)

T2 - T1 = 1/2*ma

Equating it with (1), (m2-m1)g - (m1+m2)a = 1/2*ma

a = (m2 - m1)g / (m1 + m2 + m/2)

a)

a = (545.7-485)*9.81/(545.7 + 485 + 45.5/2)

a = 0.565 m/s^2

b)

From FBD of m1: T1 - 0.485*9.81 = 0.485*0.565

T1 = 5.0319 N

c)

From FBD of m2: 0.5457*9.81 - T2 = 0.5457*0.565

m2 = 5.045 N

d) T2 - T1 = 5.045 - 5.0319 = 0.0130965 N

e)

a = (m2 - m1)g / (m1 + m2 + m/2)

Putting m = 0, a = (545.7-485)*9.81/(545.7 + 485) = 0.57773 m/s^2

T1 = 5.038049 N

T2 = 5.0380497 N

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