Equilibrium of a Rigid Body question

In summary, the conversation discusses the calculation of the weight of an object hung from a string wound on a disk with a drilled hole, in order to maintain equilibrium. The net torque must be zero, and the equation for this is set up by considering the torque of the plate with the hole, the entire disk, and the hole itself. The equation ultimately involves using sin (90 - theta) or cos(theta) to calculate the weight of the object.
  • #1
Lamoid
44
0
The solid uniform disk of radius b shown can turn freely on an axle through its center. A hole of diameter D is drilled through the disk; its center is a distance r from the axle. The weight of the material drilled out is Fwh. Find the weight of an object hung from a string wound on the disk that will hold the disk at equilibrium in the position shown.

I really can't figure this one out. I use the center as the axle but I can't find what the distance from the axle would be for the center of mass since the hole drilled in moves it from the center. Can anyone give me a hint to get me started?
 

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  • #2
Hint: A hole (no mass) can mathematically be thought of as the sum of a positive mass plus a negative mass.
 
  • #3
So I want to set up an equation where the torque from the weight is equal to the torque of the full disk minus the torque of the hole?
 
  • #4
The net torque must be zero. The torque from the "negative mass" hole will be in the opposite direction than that of a positive mass "hole".
 
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  • #5
OK so i work out the equation

(Fw)= - (r/b)(Fwh) sin theta.

Unfortunately, the answer has it being cos theta as well as there being no negative sign which makes more sense.

How I worked it out was

(Fw)(b) sin 90 = torque of the plate with hole = torque of the entire thing - torque of hole

(Fw)(b) = 0 - (Fwh)(r)(sin theta)

(Fw)= - (r/b)(Fwh) sin theta

Where did I go wrong?
 
  • #6
Lamoid said:
How I worked it out was

(Fw)(b) sin 90 = torque of the plate with hole = torque of the entire thing - torque of hole
The net torque must be zero, so that should be:
(Fw)(b) sin 90 + torque of the plate with hole = 0
or:
(Fw)(b) sin 90 + torque of the entire thing - torque of hole = 0

That will get rid of the negative sign.

(Fw)(b) = 0 - (Fwh)(r)(sin theta)

(Fw)= - (r/b)(Fwh) sin theta
Why are you using sin(theta)?
 
  • #7
Should it be sin (90 - theta)?

Edit: I mean sin (90 + theta) which would be equal to cos theta correct?
 
Last edited:
  • #8
Lamoid said:
Should it be sin (90 - theta)?

Edit: I mean sin (90 + theta) which would be equal to cos theta correct?
I'd say the first, which does equal cos(theta). :wink:
 

What is meant by equilibrium of a rigid body?

Equilibrium of a rigid body refers to a state where the body is at rest or moving with constant velocity, with no acceleration in any direction. This means that the net force and net torque acting on the body are both equal to zero.

How is equilibrium of a rigid body different from equilibrium of a particle?

Equilibrium of a particle refers to a state where the particle is at rest or moving with constant velocity. In this case, only the net force acting on the particle needs to be zero. In equilibrium of a rigid body, both the net force and net torque must be zero.

What are the conditions for a rigid body to be in equilibrium?

A rigid body is in equilibrium when the sum of all external forces acting on the body is zero, and the sum of all external torques (or moments) acting on the body is also zero. This means that the body is not experiencing any linear or rotational acceleration.

What are the types of equilibrium for a rigid body?

There are three types of equilibrium for a rigid body: stable, unstable, and neutral. A body is in stable equilibrium when a small displacement from its current position causes a restoring torque that brings it back to its original position. Unstable equilibrium occurs when a small displacement from its current position causes a torque that moves the body further away from its original position. Neutral equilibrium occurs when a small displacement from its current position does not cause any torque.

How is the concept of equilibrium of a rigid body applied in real life?

The concept of equilibrium of a rigid body is applied in various fields, such as engineering, architecture, and physics. For example, in building construction, the design and placement of supports and beams must consider the equilibrium of the structure to ensure stability and safety. In physics, the study of equilibrium of rigid bodies is essential in understanding the mechanics of objects in motion, such as the balance of forces on a seesaw or the stability of a spinning top.

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