A plum is located at coordinates (-4.0 m, 0.0 m, 6.0 m). In unit-vector notation, what is the torque about the origin on the plum due to force F whose only component is indicated by each of the following equations?
(a) Fx = 8.0 N
ô = N · m
(b) Fx = -5.0 N
ô = N · m
(c) Fz = 2.0 N
ô = N · m
(d) Fz = -3.0 N
ô = N · m
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To calculate the torque about the origin on the plum due to a force, we need to use the cross product between the position vector of the plum and the force vector. The torque can be calculated using the formula:
τ = r x F
where τ is the torque, r is the position vector, and F is the force vector.
In this case, the position vector of the plum is given as (-4.0 m, 0.0 m, 6.0 m). We can represent it as a vector r = (-4.0, 0.0, 6.0).
Let's calculate the torque for each case:
(a) Fx = 8.0 N:
To find the torque, we need to calculate the cross product between the position vector r and the force vector F = (8.0, 0.0, 0.0):
τ = (-4.0, 0.0, 6.0) x (8.0, 0.0, 0.0)
Taking the cross product of the vectors:
τ = (0.0, -48.0, 0.0) N · m
(b) Fx = -5.0 N:
Using the same process, we can calculate the cross product between r and F = (-5.0, 0.0, 0.0):
τ = (-4.0, 0.0, 6.0) x (-5.0, 0.0, 0.0)
Taking the cross product of the vectors:
τ = (0.0, 30.0, 0.0) N · m
(c) Fz = 2.0 N:
For this case, the force vector is F = (0.0, 0.0, 2.0). Calculating the cross product between r and F:
τ = (-4.0, 0.0, 6.0) x (0.0, 0.0, 2.0)
Taking the cross product of the vectors:
τ = (0.0, -12.0, 0.0) N · m
(d) Fz = -3.0 N:
Using the same process as above, we can calculate the cross product between r and F = (0.0, 0.0, -3.0):
τ = (-4.0, 0.0, 6.0) x (0.0, 0.0, -3.0)
Taking the cross product of the vectors:
τ = (0.0, 18.0, 0.0) N · m
Therefore, the torques about the origin on the plum due to the given forces are:
(a) τ = (0.0, -48.0, 0.0) N · m
(b) τ = (0.0, 30.0, 0.0) N · m
(c) τ = (0.0, -12.0, 0.0) N · m
(d) τ = (0.0, 18.0, 0.0) N · m