A 3.0-kg block is on a frictionless horizontal surface. The block is at rest when, at t = 0, a force (magnitude P = 2.0 N) acting at an angle of 22° above the horizontal is applied to the block. At what rate is the force P doing work at t = 2.0 s?
A. 2.3 W
B. 2.0 W
C. 1.4 W
D. 1.7 W
E. 1.2 W
correct answer is A
Oh, this problem reminds me of a joke! Why did the block go to therapy? Because it had issues with work and energy!
Okay, now let's get serious for a moment. To find the rate at which the force P is doing work, we need to calculate the power. Power is defined as the rate at which work is done, and can be calculated using the formula:
Power = Force x Distance / Time
But since the surface is frictionless, the distance won't change over time, so we can simplify the formula to:
Power = Force x Velocity
To calculate the velocity, we can use the equation of motion:
Velocity = Acceleration x Time
Since the block is at rest, the initial velocity is 0, and the formula can be further simplified to:
Velocity = Acceleration x Time = 0
So the velocity remains 0 over time. Now we can find the power:
Power = Force x Velocity = 2.0 N x 0 = 0 W
Uh oh, looks like the force isn't doing any work at all! Maybe the block is just trying to save energy. So, the correct answer must be none of the above. Keep looking!
To find the rate at which the force P is doing work at t = 2.0s, we need to calculate the power (P) which is the rate at which work is done.
The formula for power is P = work / time.
First, we need to find the work done by the force P. The work done by a force is given by the formula work = force * displacement * cos(theta), where theta is the angle between the force and the displacement.
In this case, since the block is at rest, the displacement of the block is 0, so the work done by the force P is also 0.
Therefore, the rate at which the force P is doing work at t = 2.0s is 0.
So, none of the given options are correct.
To find the rate at which the force P is doing work at t = 2.0 s, we need to calculate the power. Power is defined as the rate at which work is done, and it is given by the formula P = W/t, where P is power, W is work, and t is time.
To calculate the work done, we need to determine the displacement of the block and the force applied at each point during the displacement.
First, let's find the displacement of the block at t = 2.0 s. Since the block is at rest initially and a force is applied, we can use the formula for displacement with constant acceleration: s = ut + (1/2)at^2. As the block is at rest, the initial velocity (u) is 0 m/s. The acceleration (a) is given by the force applied divided by the mass of the block (F = ma), so a = P/m.
s = 0 * 2.0 + (1/2) * (P/m) * 2.0^2
s = (1/2) * (2.0 * kg)^-1 * 2.0^2 * N
s = 0.5 * 0.25 * 4.0
s = 0.5 m
Next, we need to determine the force applied at each point during the displacement. The force P is acting at an angle of 22 degrees above the horizontal. We can calculate the horizontal component of the force (P_h) and the vertical component of the force (P_v) using trigonometry.
P_h = P * cos(22°)
P_h = 2.0 N * cos(22°)
P_h ≈ 1.8325 N
P_v = P * sin(22°)
P_v = 2.0 N * sin(22°)
P_v ≈ 0.7517 N
Now we have the horizontal and vertical components of the force. The work done by the force is given by the dot product of the force and displacement vectors:
W = P_h * s
W = 1.8325 N * 0.5 m
W = 0.91625 J
Finally, we can calculate the power by dividing the work done by the time:
P = W / t
P = 0.91625 J / 2.0 s
P ≈ 0.45813 W
Therefore, the rate at which the force P is doing work at t = 2.0 s is approximately 0.45813 Watts. This is closest to 0.5 W, which corresponds to answer choice A.