Rotational Torque/Force Problem (rod around hinge)

[gapp123]

Homework Statement

A Rod of Mass "M" is attached to a two-way hinge on the floor. It is being pulled by a spring (extension spring, so force is pulling the rod to the right, causing it to move clockwise).

The hinge is frictionless.
the rod has a mass of 5kg evenly distributed
the pulling tension of the spring is 25N
The angle between the rod and floor is 25 degrees
D2 is 0.3 meters
d1 is .15 meters

How do I calculate the resulting vertical force of the rod against the ceiling? The system is in equilibrium.

I am a bit rusty with angular torque. Any help would be greatly appreciated.
[/B]

2. Homework Equations :
F=MA
Torque=NM
Hoooke's Law (I have already applied this..no need to incorporate/explain it further).
CG=(W1d1+W2d2+W3d3...)/W

3. Work Towards Problem:
I realize that since the system is in static equilibrium, the opposing forces will cancel each other out,, those forces being the tension of the spring (already solved) and the force of the rod against the ceiling. I think the answer lies in angular torque, something I am a bit rusty at.

 

[Merlin3189]

I think you need more information than you've shown, in order to solve this.

But for a start, the force from the ceiling can be considered in two parts, one perpendicular to the rod and one parallel to the rod: the total force is the resultant of these two components. The point of doing this is, that the torque on the rod is not affected by the force parallel to it, only by the perpendicular component.

To remind you, the torque due to a force is equal to the perpendicular distance from the line of action of the force, multiplied by the magnitude of the force (though you need to take .clockwise and anticlockwise into account.)

 

[gapp123]

I think you need more information than you've shown, in order to solve this.

But for a start, the force from the ceiling can be considered in two parts, one perpendicular to the rod and one parallel to the rod: the total force is the resultant of these two components. The point of doing this is, that the torque on the rod is not affected by the force parallel to it, only by the perpendicular component.

To remind you, the torque due to a force is equal to the perpendicular distance from the line of action of the force, multiplied by the magnitude of the force (though you need to take .clockwise and anticlockwise into account.)

My understanding is that the force the rod applies against the wall will be angled upwards, so trigonometry should be able to be able to dissect that force into horizontal and vertical components. I know the angle (25 degrees) of the rod against the ceiling (the angle on the right side of the rod) will be the same as the angle of the rod against the floor (angle on the left side of the rod), assuming a hinge of negligible height.

Given the information of the problem, the distance between the floor and the ceiling is 0.1902 meters.

I am struggling with the fact the pulling motion of the spring is towards the middle of the rod, not at the end. I know I need to continue with torque calculations, but this is where I am stuck.

 

[Merlin3189]

The technique with torque is to take moments about a point where you want to ignore the forces: in this case the bottom, hinge end. Provided there is no torque provide by the hinge, you can ignore pushes or pulls on the rod at that end. The only torques will be, the force of the spring on the rod and the force of the ceiling on the rod. When the rod is in equilibrium, those torques are equal and opposite.

You started with a diagram (which puts you a big step ahead of many questioners here!) Perhaps you could make another showing the forces on the rod?

 

[gapp123]

The technique with torque is to take moments about a point where you want to ignore the forces: in this case the bottom, hinge end. Provided there is no torque provide by the hinge, you can ignore pushes or pulls on the rod at that end. The only torques will be, the force of the spring on the rod and the force of the ceiling on the rod. When the rod is in equilibrium, those torques are equal and opposite.

You started with a diagram (which puts you a big step ahead of many questioners here!) Perhaps you could make another showing the forces on the rod?

Below is a diagram showing the forces of the Rod...the horizontal and vertical forces of the ceiling, and the pull of the spring

 

[haruspex]

The first thing you need to settle is the direction of the force from the ceiling. See the first section of https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/
For a refresher on torques, see
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-moments/

 

[gapp123]

I suppose I would start by stating the sum of the torques are equal to zero (with d1, d2, etc. referring to the reference point of the hinge) but I am dealing with a combination of horizontal and vertical forces...Not sure how to proceed from there.

 

[gapp123]

The first thing you need to settle is the direction of the force from the ceiling. See the first section of https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/
For a refresher on torques, see
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-moments/

So the force from the ceiling is acting at a right angle to the rod, is that correct?

 

[haruspex]

So the force from the ceiling is acting at a right angle to the rod, is that correct?

The ceiling is a flat surface. It is contacted by a corner of the rod. In the diagram at the link I posted, which object is flat and which contacts it at a corner? Where is the contact plane?

 

[gapp123]

The ceiling is a flat surface. It is contacted by a corner of the rod. In the diagram at the link I posted, which object is flat and which contacts it at a corner? Where is the contact plane?

The diagram from the link shows that the rod is the flat plane and the corner of the block is the angle. In the case of my problem, it is the other way around. Thus, I assume the force exerted by the ceiling would be downward, perpendicular to the ceiling?

 

[gapp123]

The ceiling is a flat surface. It is contacted by a corner of the rod. In the diagram at the link I posted, which object is flat and which contacts it at a corner? Where is the contact plane?

Effectively, the ceiling is the contact plane.

 

[haruspex]

The diagram from the link shows that the rod is the flat plane and the corner of the block is the angle. In the case of my problem, it is the other way around. Thus, I assume the force exerted by the ceiling would be downward, perpendicular to the ceiling?

Right.

 

[gapp123]

Ok, I think I understand now:

the sum of the torques =0= (Fspring*Vertical Distance from Hinge)-(Normal Force of Ceiling*Horizontal Distance from Hinge)-(Weight of Rod *Horizontal Distance from Hinge).

Is that right?

 

[Merlin3189]

The first thing you need to settle is the direction of the force from the ceiling. See the first section of https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/

I'm not sure how this helps. Unless I'm missing something, this reference addresses only the Normal force. It does not address the total force. OP has already correctly marked the normal force on his diagram, along with the frictional force. His problem is knowing the ratio of these two components, in order to determine the angle of the net force from the ceiling on the rod.
My advice to him has been to assume the ceiling is smooth, so that the horizontal component of force is zero.

 

[haruspex]

Ok, I think I understand now:

the sum of the torques =0= (Fspring*Vertical Distance from Hinge)-(Normal Force of Ceiling*Horizontal Distance from Hinge)-(Weight of Rod *Horizontal Distance from Hinge).

Is that right?

Yes.

 

[haruspex]

I'm not sure how this helps. Unless I'm missing something, this reference addresses only the Normal force. It does not address the total force. OP has already correctly marked the normal force on his diagram, along with the frictional force. His problem is knowing the ratio of these two components, in order to determine the angle of the net force from the ceiling on the rod.
My advice to him has been to assume the ceiling is smooth, so that the horizontal component of force is zero.

Both matter. Some do get confused about the direction of the normal force (i.e., that force which opposes interpenetration) when one object presents a corner. Indeed, with two corners in contact it is indeterminate.

Separately from that, there is the question of friction. As you noted, we have to assume there is none. In the real world, there could be a frictional component. It depends how the physical arrangement was created.
I discuss that a bit in section 3.5 at the same link.