A 200 N box is pushed up an incline 10 m long and 3 m high. The average force (parallel to the plane) is 120 N.
a.) How much work is done?
b.) What is the change in the PE of the box?
c.) What is the change in KE of the box?
d.) What is the frictional force on the box?
To solve this problem, we can use the following formulas:
a.) Work (W) = Force (F) * Distance (d) * Cosine of the angle between the force and the displacement.
b.) Change in Potential Energy (ΔPE) = Mass (m) * Gravitational Acceleration (g) * Change in height (h).
c.) Change in Kinetic Energy (ΔKE) = 1/2 * Mass (m) * Change in velocity squared (v^2).
d.) Frictional force (f) = Force (F) - Component of the force parallel to the plane.
Given:
- Force (F) = 120 N
- Distance (d) = 10 m
- Height (h) = 3 m
- Gravitational Acceleration (g) = 9.8 m/s^2
Let's calculate the solutions step by step:
a.) Work (W) = F * d * Cosθ
- In this case, the angle between the force and the displacement is the same as the angle of the incline, which is given by θ = arctan(h/d).
- θ = arctan(3/10) ≈ 16.7°
- W = 120 N * 10 m * Cos(16.7°)
- W ≈ 1200 J
b.) ΔPE = m * g * h
- Since the weight of the box is given by W = m * g, we can calculate the mass (m) by rearranging the formula to m = W / g.
- m = 200 N / 9.8 m/s^2
- m ≈ 20.4 kg
- ΔPE = 20.4 kg * 9.8 m/s^2 * 3 m
- ΔPE ≈ 600 J
c.) ΔKE = 1/2 * m * v^2
- To calculate the change in velocity (v), we need to find the initial velocity (vi) and final velocity (vf).
- Since there is no mention of any initial or final velocities, we can assume the box starts from rest and comes to rest at the end of the incline.
- Therefore, ΔKE = 0 J (There is no change in kinetic energy)
d.) Frictional force (f) = F - F_parallel
- The component of the force parallel to the plane (F_parallel) can be calculated by multiplying the force (F) by the sine of the angle between the force and the incline.
- F_parallel = F * Sinθ
- F_parallel = 120 N * Sin(16.7°)
- F_parallel ≈ 33.5 N
- Frictional force (f) = 120 N - 33.5 N
- f ≈ 86.5 N
Therefore, the answers to the questions are:
a.) The work done is approximately 1200 J.
b.) The change in potential energy (PE) of the box is approximately 600 J.
c.) The change in kinetic energy (KE) of the box is 0 J (no change).
d.) The frictional force on the box is approximately 86.5 N.
To calculate different quantities related to the scenario, we need to use the appropriate formulas and equations. Let's go through each question step by step:
a.) To find the work done, we will use the formula: Work = Force x Distance x cos(angle).
The given force is the average force, which is 120 N.
The distance is given as 10 m.
The angle in this case is the angle of the incline, and we'll call it theta.
First, we can calculate theta using the height (3 m) and the length of the incline (10 m). The angle can be found using the equation: theta = arctan(height / length).
theta = arctan(3/10)
theta ≈ 16.69 degrees
Now, we can calculate the work done:
Work = 120 N x 10 m x cos(16.69 degrees)
Note: Make sure to convert the angle to radians if your calculator is in radian mode. You can use the conversion 1 radian = 57.3 degrees.
Work ≈ 1186.4 J
Therefore, the work done is approximately 1186.4 Joules.
b.) To find the change in potential energy (PE), we use the equation: Change in PE = mass x g x height.
The mass is not given directly, but since we have the force and the work done (from the previous question), we can find it by using the equation: Work = Force x Distance = m x g x Distance.
We already know that Work = 1186.4 J and the force is 120 N. The distance is still 10 m.
1186.4 J = m x 9.8 m/s^2 x 10 m
From here, we can solve for the mass:
m ≈ 12.06 kg
Now, we can calculate the change in potential energy:
Change in PE = 12.06 kg x 9.8 m/s^2 x 3 m
Change in PE ≈ 353.6 J
Therefore, the change in potential energy of the box is approximately 353.6 Joules.
c.) The change in kinetic energy (KE) is the difference between the initial and final kinetic energy. In this case, the box starts from rest and reaches a certain velocity. Since the box starts from rest, the initial kinetic energy is zero.
To find the final kinetic energy, we need to calculate the final velocity. We can use the equation: Final velocity (vf) = Initial velocity (vi) + (2as), where "a" is the acceleration and "s" is the distance.
The incline is assumed to be frictionless, so the force parallel to the plane is the net force on the box.
Net Force = Force - Frictional force
Frictional force = Force - Net Force
In this case, the force is 120 N. The net force is the component of force parallel to the plane, which is also 120 N (from the given information). Therefore, the frictional force is zero since the net force is equal to the applied force.
Hence, there is no frictional force acting on the box.
d.) The frictional force on the box is 0 N.
To summarize the answers:
a.) The work done is approximately 1186.4 J.
b.) The change in potential energy of the box is approximately 353.6 J.
c.) The change in kinetic energy is 0 J since the box starts from rest.
d.) The frictional force on the box is 0 N.