Finding torque (vector cross product)

[vu10758]

A particle position is described by position vector r = 3i + 2j and the force vector i - 2j acts on the object.

1) Find the torque about an axis through the origin and perpendicular to the xy plane. Draw the two vectors to check your torque direction.

I used the right hand rule and found out that the torque will have a negative direction. Then I used cross product term by term.

My torque is -8 k NM.

2) Find the torque about an axis through the point x = 5m, y = 5m perpendicular to the xy plane. Draw the two vectors to check your torque direction. The force should produce a roation about (5,5) consistent with your answer.

How do I do this? The answer is 7 z Nm. Do I find the torque the same way as for part 1, and then somehow take the distance between them into consideration? How come the direction is now positive?

 

[Pyrrhus]

Ok the first question was pretty straightforward, they gave you the position vector from the origin to your force point of application. Now for the 2nd question you have another position. They give you the position vector for the other axis, and you have the position vector for your force point of application. Therefore, you can calculate the new position vector from the new axis position to your force of application. Remember is from the new axis position to your force point of application.

 

[kyle8921]

For part 1, the torque can be calculated using the formula T = r x F, where r is the position vector and F is the force vector. In this case, r = 3i + 2j and F = i - 2j. Plugging these values into the formula, we get:

T = (3i + 2j) x (i - 2j)
= 3i x i - 3i x 2j + 2j x i - 2j x (-2j)
= 0 - 6k + 2k - 0
= -8k

Since the torque is in the negative k direction, this means that the object will rotate clockwise around the origin.

For part 2, we can use the same formula but we need to take into consideration the distance between the axis and the point of rotation. The torque will be calculated using the formula T = (r - r0) x F, where r is the position vector, r0 is the position vector of the axis (in this case, (5i + 5j)) and F is the force vector. So, we have:

T = (3i + 2j - 5i - 5j) x (i - 2j)
= -2i - 3j x i + 4i - 5j x (-2j)
= 0 - 3k + 4k - 0
= 7k

Since the torque is in the positive k direction, this means that the object will rotate counterclockwise around the point (5,5). The direction of the torque can also be confirmed by using the right hand rule, where the fingers of the right hand point in the direction of the first vector (r-r0) and curl towards the direction of the second vector (F), giving a positive k direction.

In summary, to find the torque about a specific point, we use the same formula as before but we need to take into consideration the distance between the axis and the point of rotation. The direction of the torque can also be determined using the right hand rule.