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?
To find the rate at which the force P is doing work at t = 2.0 s, we need to find the dot product of the force and the velocity of the block at that particular time.
Here's how you can do it step by step:
Step 1: Calculate the acceleration of the block.
Since the block is on a frictionless surface, the only force acting on it is the force P. We can use Newton's second law, F = ma, to find the acceleration.
Given:
Mass of the block (m) = 3.0 kg
Force applied (P) = 2.0 N
Using the component of the applied force in the horizontal direction:
P_horizontal = P * cos(22°)
P_horizontal = 2.0 N * cos(22°) ≈ 1.837 N
Since there are no other forces in the horizontal direction, F_horizontal = P_horizontal = 1.837 N
Using Newton's second law:
F_horizontal = m * a_horizontal
1.837 N = 3.0 kg * a_horizontal
a_horizontal = 1.837 N / 3.0 kg ≈ 0.6123 m/s²
Step 2: Calculate the distance traveled by the block at t = 2.0 s.
We can use the equation of motion: x = x_0 + v_0 * t + 0.5 * a * t²
Given:
Initial velocity (v_0) = 0 m/s (since the block starts from rest)
Time (t) = 2.0 s
Acceleration (a) = a_horizontal = 0.6123 m/s²
Solving for x, the distance traveled by the block:
x = x_0 + v_0 * t + 0.5 * a * t²
x = 0 + 0 * 2.0 + 0.5 * 0.6123 * 2.0²
x ≈ 0.6123 m
So, at t = 2.0 s, the block has traveled approximately 0.6123 meters.
Step 3: Calculate the velocity of the block at t = 2.0 s.
Using the equation of motion: v = v_0 + a * t
Given:
Initial velocity (v_0) = 0 m/s
Time (t) = 2.0 s
Acceleration (a) = a_horizontal = 0.6123 m/s²
Solving for v, the velocity of the block:
v = v_0 + a * t
v = 0 + 0.6123 * 2.0
v ≈ 1.2246 m/s
So, at t = 2.0 s, the velocity of the block is approximately 1.2246 m/s.
Step 4: Calculate the rate at which the force P is doing work.
The work done by a force is given by the equation: W = F * d * cos(θ), where F is the force, d is the displacement, and θ is the angle between the force and the displacement.
Given:
Force (F) = magnitude of force P = 2.0 N
Displacement (d) = distance traveled by the block = 0.6123 m
Angle (θ) = angle between the force and the displacement = 22°
Using the equation for work:
W = F * d * cos(θ)
W = 2.0 N * 0.6123 m * cos(22°)
W ≈ 1.2198 J
So, the work done by the force P at t = 2.0 s is approximately 1.2198 Joules.
Step 5: Calculate the rate of work done.
The rate of work is given by the equation: power (P) = work done (W) / time (t).
Given:
Work done (W) = 1.2198 J
Time (t) = 2.0 s
Using the equation for power:
P = W / t
P = 1.2198 J / 2.0 s
P ≈ 0.6099 Watts
Therefore, the rate at which the force P is doing work at t = 2.0 s is approximately 0.6099 Watts.
To find the rate at which the force P is doing work at t = 2.0 s, we need to calculate the work done by the force P between t = 0 and t = 2.0 s, and then divide by the time interval.
First, let's calculate the work done by the force P. The work done by a force is given by the equation:
Work = Force * Displacement * cos(theta)
Where:
- Force is the magnitude of the force P (2.0 N)
- Displacement is the distance the block moves in the direction of the force
- theta is the angle between the force and the displacement
Since the block is at rest, the displacement at t = 2.0 s is given by the equation:
Displacement = 0.5 * Acceleration * Time^2
Where:
- Acceleration is the acceleration of the block due to the force P
- Time is the time interval (2.0 s - 0 s)
The acceleration of the block is given by the equation:
Acceleration = Force / Mass
Where:
- Mass is the mass of the block (3.0 kg)
Now, let's calculate the values:
Acceleration = Force / Mass = 2.0 N / 3.0 kg = 0.67 m/s^2
Displacement = 0.5 * Acceleration * Time^2 = 0.5 * 0.67 m/s^2 * (2.0 s - 0 s)^2 = 0.67 m
Work = Force * Displacement * cos(theta) = 2.0 N * 0.67 m * cos(22 degrees) ≈ 1.43 J (Joules)
Finally, we can calculate the rate at which the force P is doing work:
Rate of work = Work / Time = 1.43 J / 2.0 s ≈ 0.72 W (Watts)
Therefore, the rate at which the force P is doing work at t = 2.0 s is approximately 0.72 Watts.