Chapter 6, Problem 6SP

Suppose you wish to compare the work done by pushing a box on rollers up a ramp to the work done if you lift the box straight up to the same final height.

1. a. What work is required to lift a 178-N box (about 40 lbs) up to a table which is 0.8 m off the floor?
2. b. Let’s assume you also have a ramp available that makes an angle of 30° with the horizontal, as shown in the figure below. The ramp is 1.6 m long. The weight of the box (178 N) is due to the Earth’s pulling on the box. This 178 N is a force directed straight down. If you push it up a ramp, you are doing work against only the component of this weight along the ramp, which is 89 N, as shown in the diagram. How much work does it require to push the box up the ramp, assuming no friction?
3. c. Which situation (pushing up the ramp or lifting straight up) requires more work?
4. d. Which situation requires more force?
5. e. For which situation is the distance moved greater?
6. f. What is the change in the gravitational potential energy of the box for each situation?
7. g. What advantage, if any, is there to using the ramp? Explain.

To determine

The work required to lift the box.

Answer to Problem 6SP

The work required to lift the box is $142.4\text{J}$.

Explanation of Solution

Given info: The force over the box is $178\text{N}$ and the distance by which the box lifted is $0.8\text{m}$.

Write the expression for work done when force applied force and the displacement are in the same direction.

$W=Fd$

Here,

$W$ is the work

$F$ is the force

$d$ is the displacement of the object in the direction of the applied force

Since the gravitational force over the box is $178\text{N}$, a force equal to this value but vertically upward, must be applied to lift the box. Thus, the force is $178\text{N}$.

Substitute $178\text{N}$ for $F$ and $0.8\text{m}$ for $d$ to find the work required.

$\begin{array}{c}W=\left(178\text{N}\right)\left(0.8\text{m}\right)\\ =142.4\text{J}\end{array}$

Conclusion:

Therefore, the work required to lift the box is $142.4\text{J}$.

To determine

The amount of work required to push the box up the ramp.

Answer to Problem 6SP

The amount of work required to push the box up the ramp is $142.4\text{J}$.

Explanation of Solution

Given info: The diagram is given in the question. The distance of the ramp is $1.6\text{m}$ and the component of weight of the box parallel to the ramp is $89\text{N}$.

Since the component of the weight (gravitational force) along the ramp is $89\text{N}$, an equal amount of force must be applied to the box in the opposite direction to push it up the ramp.

Write the expression for work done when force applied force and the displacement are in the same direction.

$W=Fd$

Substitute $89\text{N}$ for $F$ and $1.6\text{m}$ for $d$ to find the work required.

$\begin{array}{c}W=\left(89\text{N}\right)\left(1.6\text{m}\right)\\ =142.4\text{J}\end{array}$

Conclusion:

Therefore, the amount of work required to push the box up the ramp is $142.4\text{J}$.

To determine

Which among the situations, pushing up the ramp or lifting straight up required more work.

Answer to Problem 6SP

The amount of work required for both pushing up the ramp and lifting straight up is the same.

Explanation of Solution

Given info: The work required to lift the box is $142.4\text{J}$ and the amount of work required to push the box up the ramp is $142.4\text{J}$.

It is obtained that the work required to lift the box straight up is $142.4\text{J}$ and that required to push the box up the ramp is $142.4\text{J}$. Therefore, the work required for both the situations are the same.

Conclusion:

Therefore, the amount of work required for both pushing up the ramp and lifting straight up is the same.

To determine

Which among the situations, pushing up the ramp or lifting straight up required more force.

Answer to Problem 6SP

Lifting the box straight up required more force than pushing it up the ramp.

Explanation of Solution

Given info: The force over the box is $178\text{N}$ and the component of weight of the box parallel to the ramp is $89\text{N}$.

Since the gravitational force over the box is $178\text{N}$, a force required to lift the box straight up is equal to this value but vertically upward. Moreover, since the component of the weight (gravitational force) along the ramp is $89\text{N}$, an equal amount of force must be applied to the box in the opposite direction to push it up the ramp. Thus, lifting the box straight up required more force than pushing it up the ramp.

Conclusion:

Therefore, lifting the box straight up required more force than pushing it up the ramp.

To determine

Which among the situations, pushing up the ramp or lifting straight up, the distance moved is greater.

Answer to Problem 6SP

The distance moved is greater in pushing the box up the ramp than that in lifting the box straight up.

Explanation of Solution

Given info: The work done for both the situations is $142.4\text{J}$ and the force required in lifting the box straight up is $178\text{N}$ and that in pushing the box up the ramp is $89\text{N}$.

By definition, the work done by a force is equal to the product of force and the displacement of the object in the direction of application of force. Here, in both the situations, the work done is same but the force required is greater in lifting the box up the ramp. The force is applied over a long distance when the box is pushed up the ramp.

Conclusion:

Therefore, the distance moved is greater in pushing the box up the ramp than that in lifting the box straight up.

To determine

The change in the gravitational potential energy of the box for lifting the box straight up and pushing it up the ramp.

Answer to Problem 6SP

The change in the gravitational potential energy of the box for lifting the box straight up and pushing it up the ramp is $142.4\text{J}$ each.

Explanation of Solution

Given info: The height of the final position of the box is $0.8\text{m}$, the gravitational force on the box is $178\text{N}$.

Since the same box is considered in both the situations, the mass $m$ is same.

Write the expression gravitational force on the object.

$F=mg$

Here,

$F$ is the gravitational force

$g$ is the acceleration due to gravity

Substitute $1785\text{N}$ for $F$ to find the value for $mg$.

$mg=178\text{N}$

Write the expression for the change in gravitational potential energy.

$\Delta PE=mg\Delta h$

Here,

$\Delta PE$ is the change in gravitational potential energy

$m$ is the mass of the object

$g$ is the acceleration due to gravity

$\Delta h$ is the change in height of the position of the object

Since in both processes, the box is moved from the initial ground level to a final height of $0.8\text{m}$. Thus, the change in height is $0.8\text{m}$

Substitute $178\text{N}$ for $mg$ and $0.8\text{m}$ for $\Delta h$ to find the change in gravitational potential energy.

$\begin{array}{c}\Delta PE=\left(178\text{N}\right)\left(0.8\text{m}\right)\\ =142.4\text{J}\end{array}$

Thus, the change in the gravitational potential energy of the box for lifting the box straight up and pushing it up the ramp is $142.4\text{J}$ each.

Conclusion:

Therefore, the change in the gravitational potential energy of the box for lifting the box straight up and pushing it up the ramp is $142.4\text{J}$ each.

To determine

The advantage of using the ramp to move the box.

Answer to Problem 6SP

When the ramp is used to move the box to the given height, less force is required than lifting it straight up to the height and hence the strength of the person doing work can be conserved using the ramp.

Explanation of Solution

As obtained in the calculations, the force required to move the box to the final position is $178\text{N}$ when lifted straight up and $89\text{N}$ when pushed up the ramp. It is clear that the required force for obtaining the same result, in the case of ramp is only nearly half of that in lifting straight up. Thus, the person who is doing the work can conserve his strength using the ramp.

Conclusion:

Therefore, when the ramp is used to move the box to the given height, less force is required than lifting it straight up to the height and hence the strength of the person doing work can be conserved using the ramp.