(a) At a certain instant, a particle-like object is acted on by a force F = (3.0 N) - (1.0 N) + (8.0 N) while the object's velocity is v = - (2.0 m/s) + (4.0 m/s) . What is the instantaneous rate at which the force does work on the object?
W
(b) At some other time, the velocity consists of only a j component. If the force is unchanged, and the instantaneous power is -12 W, what is the velocity of the object just then?
m/s
(a) To find the instantaneous rate at which the force does work on the object, we can use the dot product between the force and the velocity vectors.
The dot product of two vectors A = (A1, A2, A3) and B = (B1, B2, B3) is given by:
A · B = A1*B1 + A2*B2 + A3*B3
In this case, the force vector F = (3.0 N) - (1.0 N) + (8.0 N) can be written as F = (3.0, -1.0, 8.0) N and the velocity vector v = (-2.0 m/s) + (4.0 m/s) can be written as v = (-2.0, 4.0, 0.0) m/s.
Now, we can calculate the dot product:
F · v = (3.0 * -2.0) + (-1.0 * 4.0) + (8.0 * 0.0) = -6.0 - 4.0 + 0.0 = -10.0
Therefore, the instantaneous rate at which the force does work on the object is -10.0 Joules per second or -10.0 Watts (W).
(b) To find the velocity of the object when the power is -12 W and only has a j component, we can use the fact that power is the dot product of force and velocity vectors divided by the magnitude of the velocity vector (assuming the force is constant).
The power formula is given by:
P = (F · v) / |v|
In this case, the force vector F remains the same as in part (a), and the velocity vector v only has a j component, which is v = (0, vj, 0).
Now, we can substitute these values into the power formula:
-12 W = (F · vj) / |vj|
Since vj is the only non-zero component, the dot product simplifies to:
Fj * vj = -12 * |vj|
Now, let's find the value of vj:
Fj * vj = -12 * |vj|
-1.0 * vj = -12 * |vj|
Dividing both sides by -1.0:
vj = 12 * |vj|
Since the velocity is a scalar value, the magnitude of vj is equal to |vj|.
Therefore:
vj = 12 * vj
Dividing both sides by vj:
1 = 12
This equation has no solutions, which means there is no velocity that satisfies the condition of having only a j component and a power of -12 W.