Compute the torque about the origin of the gravitational force F = −mg acting on a particle of mass m located at r = x + y. (Use the following as necessary: m, g, x, and y). Torque = F (cross) r
Please help
If F= - (m•g)•j and r = (x) •I + (y) •j.
τ= - (m•g•x) [i x j] – (m•g•y)[j xj] = (m•g•x)•k.
i, j, k are the unit vectors of Cartesian coordinate system.
To compute the torque about the origin of the gravitational force F = -mg acting on a particle of mass m located at r = x + y, we will use the formula Torque = F (cross) r.
First, let's calculate the cross product of the force F and the position vector r. The cross product of two vectors can be calculated using the determinant:
F x r = |i j k|
|Fx Fy Fz|
|Rx Ry Rz|
Since the problem only gives us the force F = -mg in the y-direction, with Fy = -mg, we can set Fx = Fz = 0.
Then, the position vector is given as r = x + y, so we can write Rx = x, Ry = y, and Rz = 0.
Plugging the values into the determinant, we have:
F x r = |i j k|
|0 -mg 0 |
|x y 0|
Using the first row as the reference, we can calculate the determinant:
F x r = (0 * 0 - (-mg) * y) * i - (0 * 0 - x * 0) * j + (0 * y - x * (-mg)) * k
= mgx * k
Thus, the torque about the origin of the gravitational force F = -mg acting on a particle of mass m located at r = x + y is given by:
Torque = F x r = mgx * k