A 10 kg box starts from rest and is then pulled horizontally 27 meters across the ground by a rope. The rope has a force of 100N applied to it and forms an angle of 50 degree to the ground. The coefficient of kinetic friction is 0.50 for this surface.
(a)How much work is done by the applied force?
(b)How much work is done by friction?
(c) How much work is done by gravity?
(d)How much work is done by the normal force?
(e)What;s the total work done on the box?
(f)what is the change in potential energy of the box?
(i)What is the change in potential energy of the box?
(j)how long did it take for the box to travel 18 meters?
(k) what is the total power?
a) work=100cosTheta*distance
b)frictionwork=mu(mg-100sinTheta)*distance
To solve the given questions, we can use the work-energy principle. The work done is calculated using the formula:
Work = Force x Distance x cosine(angle)
Let's go through each question one by one:
(a) To calculate the work done by the applied force:
Work = Applied Force x Distance x cosine(angle)
= 100N x 27m x cos(50°)
≈ 1,894.42 Joules
Therefore, the work done by the applied force is approximately 1,894.42 Joules.
(b) To calculate the work done by friction:
Frictional Force = Normal Force x Coefficient of Kinetic Friction
= (mass x gravity) x coefficient of kinetic friction
= (10 kg x 9.8 m/s^2) x 0.50
= 49N
Work = Frictional Force x Distance x cosine(180°) [cosine of the angle between the force and displacement]
= 49N x 27m x cos(180°)
= -1,323.03 Joules
Therefore, the work done by friction is approximately -1,323.03 Joules.
Note: The negative sign indicates that the direction of work done by friction is opposite to the direction of displacement.
(c) To calculate the work done by gravity:
Work = Weight x Distance x cosine(270°) [cosine of the angle between the force and displacement]
= Weight x Distance x 0
= 0
Therefore, the work done by gravity is zero.
(d) To calculate the work done by the normal force:
As the displacement is horizontal, the work done by the normal force is also zero.
(e) To calculate the total work done on the box:
Total Work = Work done by applied force + Work done by friction + Work done by gravity + Work done by normal force
= 1,894.42 - 1,323.03 + 0 + 0
≈ 571.39 Joules
Therefore, the total work done on the box is approximately 571.39 Joules.
(f) To calculate the change in potential energy of the box:
Change in potential energy = mgh
= mass x gravity x height
= 10 kg x 9.8 m/s^2 x 0
= 0
Therefore, the change in potential energy of the box is zero.
(i) The change in potential energy of the box remains zero.
(j) To calculate the time it took to travel 18 meters:
The work done by the applied force is equal to the change in kinetic energy:
Work done by applied force = Change in kinetic energy
100N x 18m x cos(50°) = (1/2) x mass x (final velocity)^2 - (1/2) x mass x (initial velocity)^2
Using m = 10 kg
9,486 J = (1/2) x 10 kg x (final velocity)^2 - (1/2) x 10 kg x 0^2
final velocity^2 = 1,897.2
final velocity = √(1,897.2)
final velocity ≈ 43.57 m/s
Using the equation v = u + at, where u = 0 m/s, a = 0 m/s^2, and v = 43.57 m/s:
43.57 = 0 + 0 x t
t = 0 / 0 (same initial and final velocities)
t = 0 seconds
Therefore, it took 0 seconds for the box to travel 18 meters.
(k) To calculate the total power:
Power = Work / Time
As the time taken is 0 seconds (instantaneous), the total power will be undefined (since dividing by zero is undefined).
Hence, the total power is undefined.
To answer the questions, we need to analyze the forces acting on the box and calculate the work done by each force.
(a) The work done by the applied force can be found using the formula: work = force × distance × cosθ, where θ is the angle between the force and the displacement.
Given:
Applied force = 100 N
Distance = 27 m
Angle = 50 degrees
Plugging in the values, we have: work = 100 N × 27 m × cos(50°) = 100 N × 27 m × 0.6428 = 1739.56 J
Therefore, the work done by the applied force is 1739.56 J.
(b) The work done by friction can be found using the formula: work = force of friction × distance.
To find the force of friction, we can use the equation: force of friction = coefficient of kinetic friction × normal force.
Given:
Coefficient of kinetic friction = 0.50
Normal force = mass × gravity
Since the box is on a horizontal surface and not accelerating vertically, the normal force equals the weight of the box, which is given by: weight = mass × gravity.
Given:
Mass = 10 kg
Gravity = 9.8 m/s^2
So, the normal force = 10 kg × 9.8 m/s^2 = 98 N
Now, we can find the force of friction: force of friction = 0.50 × 98 N = 49 N
Finally, the work done by friction can be calculated: work = 49 N × 27 m = 1323 J
Therefore, the work done by friction is 1323 J.
(c) The work done by gravity can be calculated using the formula: work = force × distance × cosθ.
Since the displacement is horizontal, the angle between the force of gravity and the displacement is 90 degrees, and cos90° = 0.
Thus, the work done by gravity is 0 J.
(d) The work done by the normal force is also zero, as the normal force acts perpendicular to the displacement, resulting in an angle of 90 degrees and cos90° = 0.
Therefore, the work done by the normal force is also 0 J.
(e) The total work done on the box is the sum of the work done by each force.
Total work done = work done by applied force + work done by friction + work done by gravity + work done by normal force
Total work done = 1739.56 J + 1323 J + 0 J + 0 J = 3062.56 J
Therefore, the total work done on the box is 3062.56 J.
(f) The change in potential energy can be found using the formula: change in potential energy = mass × gravity × change in height, assuming no change in height.
Since the box moves horizontally, there is no change in height, so the change in potential energy is 0.
Therefore, the change in potential energy of the box is 0 J.
(i) Since the height does not change, the change in potential energy of the box is 0 J.
(j) To find the time it took for the box to travel 18 meters, we need to calculate the average speed first.
Average speed = distance / time
Given:
Distance = 18 m
Average speed = 18 m / time
From the problem statement, we know that the box starts from rest, so the final speed is zero. Since the motion is uniform, we can use the equation of motion: v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
Since the box starts from rest, the initial velocity u is 0 and the acceleration a is also 0 (no acceleration horizontally). Therefore, the equation becomes: v^2 = 0^2 + 2(0)(18) = 0
This means the box does not have a final velocity after traveling 18 meters.
Without a velocity, we cannot calculate the time using average speed because it requires dividing by time, which would result in division by zero.
So, we cannot determine the time it took for the box to travel 18 meters based on the given information.
(k) The power can be calculated using the formula: power = work done / time.
Given:
Total work done = 3062.56 J
To calculate power, we need the time it took for the box to be pulled.
Unfortunately, the time is not provided in the given information, so we cannot calculate the total power at this time.