Rigid Body Rotation: Calculating Load Supported by Pivot

In summary, the conversation discussed solving for the load supported by a pivot, which was found to be Mg/4. The steps involved figuring out the acceleration of the center of mass and applying Newton's 2nd law for translation, considering the forces acting on the wood, and calculating the net force at the pivot. The final solution was N = 1/4mg.
  • #1
wolf party
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Homework Statement



imagine a thin length of wood supported by two supports, one at either end. One support dissapears and the other turns into a pivot (as gravity acts on the wood). Show that the load supported by the pivot is Mg/4.


The Attempt at a Solution



i don't know where to start due to my confusion on forces with rotation. i know the plank is acted on by gravity at its CM, and its moment of inertia about its CM = I=1/12*M*l^2.
I know there is a reacton force at the Pivot, and this is the force i need to calculate, but i don't know what to consider first, a nudge in the right direction would be helpful!
 
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  • #2
Start by figuring out the acceleration of the center of mass. Hint: Apply Newton's 2nd law to the rotation.
 
  • #3
F=ma -> TAU = I*ALPHA

TAU = (1/12*M*l*l)*ALPHA

so ALPHA = TAU/(1/12*M*l*l)

so a = (TAU/(1/12*M*l*l)) * (0.5*l) ? where a=ALPHA*(0.5l)
 
  • #4
wolf party said:
TAU = (1/12*M*l*l)*ALPHA
(1) It pivots about one end, not the center of mass, so correct the rotational inertia.
(2) What's the torque about the pivot?
 
  • #5
TAU = (1/3*M*L*L)*ALPHA

the torque about the pivot is mgl/2 ?
 
  • #6
Good. Keep going.
 
  • #7
so the acceleration due to gravity is a = 3*TAU/2*ML at the CM
and the torque about the pivot is mgl/2

so m*a = TAU*(1/2*l) ?
 
  • #8
wolf party said:
so the acceleration due to gravity is a = 3*TAU/2*ML at the CM
and the torque about the pivot is mgl/2
Good. So what's the acceleration of the center of mass? (Don't call it "acceleration due to gravity".)

Once you have the acceleration of the center of mass, apply Newton's 2nd law for translation. What forces act on the wood?
 
  • #9
is the acceleration ofthe CM just 3TAU/2mL again
the reacton force at the pivot and mg at the CM are the forces on the wood

R = 0 because mgl/2 (where l = 0) = 0

therfore ma=mg ?
 
  • #10
acceleration at CM = 3/4*g
 
  • #11
wolf party said:
is the acceleration ofthe CM just 3TAU/2mL again
But you know the torque, so eliminate Tau from this expression.
the reacton force at the pivot and mg at the CM are the forces on the wood
Good.
R = 0 because mgl/2 (where l = 0) = 0

therfore ma=mg ?
No. Apply Newton's 2nd law. What's net force? The acceleration?
 
  • #12
wolf party said:
acceleration at CM = 3/4*g
Good!
 
  • #13
the force at the CM=3/4*mg

Net force at pivot = force at CM ?
 
  • #14
wolf party said:
the force at the CM=3/4*mg
That's the "ma" part of "Fnet = ma". But what's Fnet?
Net force at pivot = force at CM ?
No. Two forces act on the wood (you named them earlier). Combine them to get the net force.
 
  • #15
Fnet = mg + mgl/2
?
 
  • #16
FNet = N - mg = 3/4mg

therfore N = 1/4mg ?
 
  • #17
wolf party said:
FNet = N - mg = 3/4mg
Careful. As written, this implies that N = 7/4mg. But the acceleration is downward, thus negative: N - mg = -3/4mg.

therfore N = 1/4mg ?
Yep.
 

FAQ: Rigid Body Rotation: Calculating Load Supported by Pivot

1. What is rigid body rotation?

Rigid body rotation is a type of motion in which an object moves in a circular or rotational path without any deformation or change in shape. This type of motion is often observed in objects rotating around a fixed pivot point.

2. How is the load supported by a pivot calculated in rigid body rotation?

The load supported by a pivot in rigid body rotation can be calculated using the principles of torque and equilibrium. The equation for calculating the load is T = F x d, where T is the torque, F is the force applied, and d is the perpendicular distance from the pivot to the line of action of the force.

3. What factors affect the load supported by a pivot in rigid body rotation?

The load supported by a pivot in rigid body rotation is affected by several factors, including the magnitude and direction of the applied force, the distance of the force from the pivot point, and the mass and distribution of mass of the rotating object.

4. Can the load supported by a pivot in rigid body rotation be increased?

Yes, the load supported by a pivot in rigid body rotation can be increased by increasing the magnitude of the applied force or by moving the force closer to the pivot point. However, there are limits to how much the load can be increased before the object reaches its breaking point.

5. How is rigid body rotation different from other types of motion?

Rigid body rotation is different from other types of motion, such as translational motion, because it involves a rotational movement around a fixed point without any deformation or change in shape of the object. In contrast, translational motion involves movement in a straight line without any rotation.

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