Question

Recall your experimental setup from Lab 05A: a constant force was applied to a disc by attaching a mass to a light string wrapped around a mass-less pulley and hanging the mass over the edge of the apparatus. In the lab, you used energy conservation arguments to derive an expression for the angular velocity of the disc after the mass had fallen a distance x. Your goal now is to use kinematics and dynamics to confirm your expression.

Use the following symbols throughout this question:

m is the mass of the hanging mass,

M is the mass of the disc,

r is the radius of the pulley,

R is the radius of the disc,

x is the distance the mass has fallen, and

g is the acceleration due to gravity.

What is the linear acceleration of the mass after it has fallen a distance x? Answer symbolically. HINT: Sketch the situation and construct a free-body diagram. Consider all force and torque relationships present in the setup.  

a=________ m/s^2

What is the angular velocity of the disc at the same moment? Consider kinematic relationships and the connection between linear and angular quantities when answering this question. Answer symbolically.

w=________ rad/s

Answer 1

the free body diagram is

using equilibrium is

T - mg = - ma

T = mg - ma

= T r

and = I , = a/r

T r = MR² / 2 x ( a/r )

T = M R² a / 2 r²

mg - ma = M R² a / 2 r²

mg = ma + M R² a / 2 r²

mg = a ( m + M R² / 2 r² )

a = mg / ( m + M R² / 2 r² )

the linear acceleration of the mass is a = 2mgr² / (2mr² + MR²) m/sec²

same way at equilibrium

1/2 MR² / 2 + 1/2 m v² = m g x

v = r

1/2 MR² / 2 + 1/2 m ( r)² = m g x

² ( MR²/4+ mr²/2) = m g x

² = m g x / ( MR²/4+ mr²/2)

= 4mgx / MR² + 2mr²

the angular velocity is = (4mgx / MR² + 2mr²)^1/2   rad/sec