physics help needed!!! Rotational motion?

Question

1.) the uniform door to a bank vault is 2.0 m high and 1.0 m wide and is supported by frictionless hindges. To determine the door's mass, a person pushes against its free end with a constant force of 250 N always directed perpindicularly to its plane and finds that 1.5 s is needed to turn the door through 90 degrees. What is the doors mass?

2.) An object of mass (m) on a frictionless table is attatched to the end of a string that passes through a frictionless hole in the table. The object is moving in a circle radius (R) with the angular speed omega (w) when the string is pulled through the hole until the object's orbital radius is R/2
a.) What is the new angular speed if the object?
b.) How much work had to be done to pull the string?

Answer 1

You just need to use torques :

Remember that the Force makes the door turns, so well, I have only a doubt, We all know that the moment of inertia of a line for example is :

I = 1/3*M*L^2

Using

I = integrate(x^2.dm)

I got this answer : 1/3*M*L^2 for a line

F.L = I*(alpha)........ (*)

alpha = angular acceleration

The angular acceleration can be calculated with circular motion, the door turns 90º, the initial angular velocity is cero, and the time is 1.5 s.

Wo = 0

t = 1.5 s

theta = pi/2

pi/2 = 1/2(alpha)*1.5^2

alpha = pi*1.5^2

replacing it on (*)

250.1 = pi*1.5^2*(M*1/3)

M = 333 kg

2) For this you just need to apply the conservation of angular momentum :

L = constant, and you know that L = I.W

then :

I.Wo = I.Wf

Wo = initial angular velocity

Wf = final angular velocity

for the disc : I = MR^2 / 2

MR^2/ 2*W = MR^2/4*W'

W' = 2W rad / s

Hope that helped