Question

A solid wood door 1.00 m wide and 2.00 m high is hinged along one side and has a total mass of 40.0 kg . Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass 0.300 kg , traveling perpendicular to the door at 14.0 m/s just before impact.

1.Find the final angular speed of the door.

2.Does the mud make a significant contribution to the moment of inertia?

Answer 1

w = width of door = 1 m

M = mass = 40 kg

moment of inertia of door = I = (1/3) Mw³ = (1/3) (40) (1)³ = 13.33 kgm²

W = angular speed

intial angular momentum of mud = final Total angular momentum of mud + door

m v r = I W

0.3 x 14 x 0.5 = 13.33 W

W = 0.16 rad/s

the mass of mud is very small as compared to the door, hence the contridution is insignificant

Answer 2

The parallel-axis theorem states that the moment of inertia of the body about any other axis parallel to the original one is equal to the sum of the moment of inertia through the center of mass and the moment of inertia about the parallel axis from a distance d.

Here, is the moment of inertia through the center of mass, is the mass of the body, and is the distance from the center of mass of the body to point P on the first axis.

The moment of inertia of the rectangular door about an axis passing through one side is,

Here, is the mass of the door and is the length of the door.

The moment of inertia of the mud through the center of mass is,

Here, is the mass of the mud.

Thus, the moment of inertia of the door-mud system is,

Substitute for, for, and 1.00 m for.

From the law of conservation of angular momentum, the total initial angular momentum of the door-mud system is equal to the total final angular momentum of the door-mud system.

Here, is the linear speed of the mud and is the angular speed of the door.

Thus, the angular speed of the door is,

Substitute for, for, for, and 1.00 m for.

Therefore, the angular speed of the door is.

No. The mud increases the moment of inertia of the system by only a small amount. So, the mud does not make a large (or a significant) contribution to the moment of inertia.