At a certain instant, a particle-like object is acted on by a force F = (4.0 N) - (2.0 N) + (9.0 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?
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?
To find the instantaneous rate at which the force does work on the object, we can use the dot product between the force vector and the velocity vector. The dot product is calculated by multiplying corresponding components of the vectors and summing them up.
Given:
Force vector F = (4.0 N) - (2.0 N) + (9.0 N)
Velocity vector v = (-2.0 m/s) + (4.0 m/s)
Step 1: Calculate the dot product of the force and velocity vectors.
F · v = (4.0 N)(-2.0 m/s) + (-2.0 N)(4.0 m/s) + (9.0 N)(0 m/s)
= -8.0 N m/s + -8.0 N m/s + 0
= -16.0 N m/s
Therefore, the instantaneous rate at which the force does work on the object is -16.0 N m/s.
Now, let's move on to the second part of the question.
Given:
Force vector F = (4.0 N) - (2.0 N) + (9.0 N)
Instantaneous power = -12 W
Step 2: Calculate the magnitude of the force using the given power.
Power = Force · velocity
-12 W = F · v
Since the velocity vector only has a j component, we can write the dot product as:
-12 W = Fj * vj
-12 W = F * v
Now we can solve for the magnitude of the force:
F * v = -12 W
F = (-12 W) / v
F = -12 N/(vj)
Step 3: Substitute the force magnitude into the force vector equation and solve for vj.
F = (4.0 N) - (2.0 N) + (9.0 N)
-12 N/(vj) = 4.0 N - 2.0 N + 9.0 N
-12 N/(vj) = 11.0 N
Solving for vj:
vj = -12 N / (11.0 N)
vj = -1.09 m/s
Therefore, the velocity of the object at that instant is -1.09 m/s in the j direction.
To find the instantaneous rate at which the force does work on the object, you can use the formula:
Power = Force ⋅ Velocity
In this case, the force acting on the object is given as F = (4.0 N, -2.0 N, 9.0 N) and the velocity is given as v = (-2.0 m/s, 4.0 m/s, 0).
To calculate the power, you need to take the dot product of the force and velocity vectors:
Power = (Force ⋅ Velocity)
Dot Product = (4.0 N) * (-2.0 m/s) + (-2.0 N) * (4.0 m/s) + (9.0 N) * (0)
Dot Product = -8 N⋅m/s + (-8 N⋅m/s) + 0
Dot Product = -16 N⋅m/s
Therefore, the instantaneous rate at which the force does work on the object is -16 Watts (W).
Next, to find the velocity of the object at the other time when it consists of only a j-component and the instantaneous power is -12 W, you can rearrange the power formula:
Power = Force ⋅ Velocity
Using the given information, the power is -12 W and the force remains unchanged (F = (4.0 N, -2.0 N, 9.0 N)).
Since the velocity consists of only a j-component, we can write it as v = (0, vj, 0), where v is the unknown velocity we need to find.
Again, taking the dot product of the force and velocity vectors, we get:
(-12 W) = (4.0 N) * (0) + (-2.0 N) * (vj) + (9.0 N) * (0)
Simplifying the equation gives:
- 2.0 N * vj = -12 W
Dividing both sides by -2.0 N gives:
vj = 6 m/s
Therefore, the velocity of the object at that time is v = (0, 6 m/s, 0).