Question

1. The pulley has a radius of 2.70 m and a moment of inertia of 39.0 kg·m2. The hanging mass is 4.20 kg and it exerts a force tangent to the edge of the pulley. What is the angular acceleration of the pulley? (answer needed in rad/s^2

2. A string is wound tight around the spindle of a top, and then pulled to spin the top. While it is pulled, the string exerts a constant torque of 0.150 N·m on the top. In the first 0.220 s of its motion, the top reaches an angular velocity of 12.0 rad/s. What is the moment of inertia of the top? (answer needed in kg m^2)

3. A net torque of 21 N·m acts on a pulley and causes an angular acceleration of 17 rad/s². Find the pulley’s moment of inertia. (answer needed in kg m^2)

4. For a wheel with moment of inertia 48 kg·m², what net torque is required to produce an angular acceleration of 3.4 rad/s²?(answer needed in N*m)

Answer 1

1) r = radius = 2.70 m

I = moment of interia = 39 kg m²

m = mass = 4.20 kg

$\alpha$ = angular acceleration

here mass will accelerate downward with acceleration a .T ( tension ) in string will be in upward direction.

so net force = mg - T

ma = mg - T

T = mg - ma --------------------------1

net torque = T*r ------------------2 ( torque = F(force)*r)

also torque = I* $\alpha$    ----------------3

from 2 &3:

T*r = I* $\alpha$ = I*a/ r [ a = $\alpha$ *r ,    $\alpha$ = a/r]

T = I*a / r²   --------------------4

from 1 and 4:

mg - ma = I*a / r²

a ( linear acceleration) = mg / [ m+ ( I/r²)]

= 4.20*9.8 / [ 4.20 + ( 39 / (2.70)² ) ]

= 4.31 m/s²

$\alpha$ = a/ r = 4.31 / 2.70 = 1.596 rad/s²