In unit-vector notation, what is the torque about the origin on a particle located at coordinates (0, -4.90 m, 4.04 m) due to (a) force 1 with components F1x = 2.67 N and F1y = F1z = 0, and (b) force 2 with components F2x = 0, F2y = 1.23 N, F2z = 5.41 N?
To find the torque about the origin on a particle, we can use the cross product between the position vector and the force vector. In unit-vector notation, the cross product formula is:
Torque = r x F
where r is the position vector and F is the force vector.
(a) To find the torque due to force 1, first, we need to express the position vector and force vector in unit-vector notation.
Position vector:
r = (0)i - (4.90)j + (4.04)k
Force vector:
F1 = (2.67)i + (0)j + (0)k
Now we can calculate the cross product using the following formula:
r x F1 = (r_y * F1_z - r_z * F1_y)i + (-r_x * F1_z + r_z * F1_x)j + (r_x * F1_y - r_y * F1_x)k
Plugging in the values:
r x F1 = (-4.90 * 0 - 4.04 * 0)i + (0 * 0 - 4.04 * 2.67)j + (4.90 * 2.67 - (-4.90 * 0))k
Simplifying:
r x F1 = 0i + (0 - 10.798)j + (13.053)k
Therefore, the torque about the origin due to force 1 is:
Torque1 = -10.798j + 13.053k
(b) To find the torque due to force 2, we use the same process:
Position vector:
r = (0)i - (4.90)j + (4.04)k
Force vector:
F2 = (0)i + (1.23)j + (5.41)k
Calculating the cross product:
r x F2 = (r_y * F2_z - r_z * F2_y)i + (-r_x * F2_z + r_z * F2_x)j + (r_x * F2_y - r_y * F2_x)k
Plugging in the values:
r x F2 = (-4.90 * 5.41 - 4.04 * 1.23)i + (0 * 5.41 - 4.04 * 0)j + (0 * 1.23 - (-4.90 * 0))k
Simplifying:
r x F2 = (-26.574 - 4.975)i + (0)j + (0)k
Therefore, the torque about the origin due to force 2 is:
Torque2 = -31.549i
Thus, the torque about the origin on the particle due to force 1 is -10.798j + 13.053k and due to force 2 is -31.549i in unit-vector notation.