Calculating Torque with Vectors at an Angle

[lukatwo]

Homework Statement

So I was wondering how do we account for vectors that are at a certain angle. The problem that I'm having with the picture is: if I was calculating torque from point B, how would I account for the vector G.

Homework Equations

The Attempt at a Solution

I've tried taking the component that's vertical to the slope, but in that case what is the lever arm? Is it where the vector G crosses the slope(point D)?

 

[SammyS]

Homework Statement

So I was wondering how do we account for vectors that are at a certain angle. The problem that I'm having with the picture is: if I was calculating torque from point B, how would I account for the vector G.

Homework Equations

The Attempt at a Solution

I've tried taking the component that's vertical to the slope, but in that case what is the lever arm? Is it where the vector G crosses the slope(point D)?

Try taking the component that's perpendicular to the ramp (incline).

Do you know about the line of action for a force -- in this case force G ?

 

[lukatwo]

I've tried taking the perpendicular component, but not sure what the lever arm is. I'm not sure how to determine the line of action.

 

[electronicsguy]

I've tried taking the perpendicular component, but not sure what the lever arm is. I'm not sure how to determine the line of action.

By extending the force T, you can see that it will contact point D. So it is like you're applying the force T directly on point D and the lever arm will be BD. Then take the perpendicular component of force T relative to the slope.

You can always check.
If we assume that BT is perpendicular to force T (note: BT is the dist. bet. B and T)
From the figure, the τ = T*(BT) . But, BT = BDsin(90-α). Therefore τ = T*(BDsin(90-α)) [1].
Returning to your problem, τ = Txr where r is the lever arm. The perpendicular component of T is Tsin(90-α). Therefore τ = Tsin(90-α)*r = T*(BDsin(90-α)) from [1]. Therefore r = BD.

 

[HalfLostAndSearching]

I would approach this problem by first understanding the concept of torque and its calculation. Torque is a measure of the force that causes an object to rotate around an axis. It is calculated by multiplying the force applied by the distance from the axis of rotation. In this case, the force is represented by the vector G and the distance is the lever arm.

To account for vectors at an angle, we can use the concept of vector decomposition. This means breaking down the vector into its horizontal and vertical components. In this case, we can break down vector G into its horizontal and vertical components by using trigonometric functions. The vertical component of G will contribute to the torque, while the horizontal component will not. The lever arm will be the distance from the axis of rotation to the point where the vertical component of G intersects the slope, which in this case is point D.

To calculate the torque at point B, we can use the formula: τ = F x d, where F is the vertical component of vector G and d is the lever arm. This will give us the magnitude of the torque. To determine the direction of the torque, we can use the right-hand rule, where the direction of the torque is perpendicular to both the force vector and the lever arm.

In summary, to account for vectors at an angle when calculating torque, we can use vector decomposition to find the vertical component of the force and use the lever arm as the distance from the axis of rotation to the point where the vertical component intersects the slope. I hope this helps with your problem.