# How to calculate torque with cross product?

• mohemoto
In summary, a rod with one end at the origin and one end at point P (1m, 2m, 2m) is acted upon by a force F = (3i+2j-1k) N at point P. To find the torque about the origin due to F, use the cross product formula and the definition of cross product to obtain the components of torque in the x, y, and z directions, taking into account the direction of each force and the axis of rotation. By convention, counterclockwise moments are positive.
mohemoto

## Homework Statement

A rod has one end at the origin and one end at the point P whose coordinates are (1m, 2m, 2m). A force F = (3i+2j-1k) N acts on the rod at the point P. What is the torque about the origin due to F?

torque = F x r

## The Attempt at a Solution

I'm not sure what to multiply by what. Are there any suggestions?

Well what is the vector OP which is the same as your vector r?

Use the definition of cross product. (If it's still unclear, do an internet search on "determinant of a matrix.")

$$\vec a \times \vec b = \left| \begin{array}{ccc} \hat \imath & \hat \jmath & \hat k \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{array} \right|$$

Since the force is already given in mutually perpendicular directions x , y, and z, (i, j, and k unit vectors), then use Torque = force times perpendicular distance from line of action of force to the axis about which you are summing moments.
T_x = F_y(z) + F_z(y)
T_y = F_z(x) + F_x(z)
T_z = F_y(x) + F_x(y)

Please watch plus and minus signs. By convention, counterclockwise moments about an axis are taken as positive (x axis points right positive, y-axis points up positive, and z axis points out of plane toward you as positive).

To calculate torque using the cross product, you will need to take the vector cross product of the force vector and the position vector from the origin to point P. This can be written as T = F x r, where T is the torque, F is the force vector, and r is the position vector. In this case, the position vector can be written as r = (1i+2j+2k) m. The cross product of F and r can be calculated using the determinant method or the right-hand rule. Once you have the cross product, the magnitude of the resulting vector will give you the torque about the origin.

## 1. What is torque and why is it important?

Torque is a measure of the force that can cause an object to rotate around an axis. It is important because it helps us understand how forces affect the motion of objects, and is crucial in designing and analyzing machines and structures.

## 2. How is torque calculated using the cross product?

Torque can be calculated using the cross product of two vectors: the vector from the point of rotation to the point where the force is applied, and the vector representing the direction and magnitude of the force. The magnitude of torque is equal to the product of the magnitude of these two vectors multiplied by the sine of the angle between them.

## 3. What is the relationship between torque and angular acceleration?

Torque is directly proportional to the angular acceleration of an object. This means that a greater torque will result in a larger angular acceleration, and vice versa. This relationship is described by Newton's Second Law for rotational motion, which states that the net torque is equal to the moment of inertia multiplied by the angular acceleration.

## 4. Can torque be negative?

Yes, torque can be negative. This occurs when the force is applied in a direction opposite to the direction of rotation. Negative torque results in the object slowing down or rotating in the opposite direction.

## 5. What are some real-life examples of torque?

Some examples of torque in everyday life include using a wrench to loosen a bolt, twisting a doorknob to open a door, and using a steering wheel to turn a car. In industrial settings, torque is important in machines such as turbines, engines, and gearboxes.

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