Chapter 6, Problem 6SP

(a)

To determine

The work required to lift the box.

Answer to Problem 6SP

The work required to lift the box is $100\text{ J}$.

Explanation of Solution

Given info: The weight of the box is $100\text{N}$ and the distance by which the box has to be lifted is $1\text{m}$.

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

$W=Fd$ (1)

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 weight of the box is $100\text{N}$, a force equal to this value but vertically upward, must be applied to lift the box. Thus, the force is $100\text{N}$.

Substitute $100\text{N}$ for $F$ and $1\text{m}$ for $d$ in the above equation to find the work required.

$\begin{array}{c}W=\left(100\text{N}\right)\left(1\text{m}\right)\\ =100\text{ J}\end{array}$

Conclusion:

Therefore, the work required to lift the box is $100\text{ J}$.

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 $100\text{J}$.

Explanation of Solution

Given info: The length of the ramp is $2\text{m}$ and the component of weight of the box parallel to the ramp is $50\text{N}$.

Since the component of the weight (gravitational force) along the ramp is $50\text{N}$, an equal amount of force must be applied to the box in the opposite direction to push it up the ramp.

Substitute $50\text{N}$ for $F$ and $2\text{m}$ for $d$ in equation (1) to find the work required.

$\begin{array}{c}W=\left(50\text{N}\right)\left(2\text{m}\right)\\ =100\text{J}\end{array}$

Conclusion:

Therefore, the amount of work required to push the box up the ramp is $100\text{J}$.

(c)

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 are the same.

Explanation of Solution

The word work defines an activity involving mental or physical health but in physics it is defined as the force causing movement or displacement of an object. The SI unit of work is Joule.

If the force and distance moved are in same direction, then the work is the applied force multiplied by the distance object moves. The work done by a given force is the product of the component of the force along the line of motion of the object multiplied by the distance that the object moves under the influence of the force.

From part (a) it is obtained that the work required to lift the box straight up is $100\text{J}$ and from part (b) that required to push the box up the ramp is also $100\text{J}$. This implies 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 are the same.

(d)

To determine

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

Answer to Problem 6SP

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

Explanation of Solution

The capacity to make a physical change or to do work is known as force. The work done by a given force is the product of the component of the force along the line of motion of the object multiplied by the distance that the object moves under the influence of the force.

Since the gravitational force over the box is $100\text{N}$, the force required to lift the box straight up is equal to this value but it direction must be vertically upward. Also since the component of the weight (gravitational force) along the ramp is $50\text{N}$, an equal amount of force must be applied to the box in the opposite direction to push it up the ramp.

Conclusion:

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

(e)

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

The word work defines an activity involving mental or physical health but in physics it is defined as the force causing movement or displacement of an object.

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. If work done in two situations are equal, the situation in which more force is applied will have smaller displacement of the object on which the force is acting.

In both the given situations, the work done is same. From part (d) the force required is greater in lifting the box straight up. This implies 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.

(f)

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 $100\text{J}$ each.

Explanation of Solution

Given info: The height of the final position of the box is $1\text{m}$, the gravitational force on the box is $100\text{N}$.

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

Write the expression gravitational force on the object.

$F=mg$

Here,

$F$ is the gravitational force

$m$ is the mass of the object

$g$ is the acceleration due to gravity

Substitute $100\text{N}$ for $F$ in the above equation to find the value of $mg$.

$mg=100\text{N}$

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

$\Delta h$ 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 $1\text{m}$, the change in height is $1\text{m}$ in both cases.

Substitute $100\text{N}$ for $mg$ and $1\text{m}$ for $\Delta h$ in the above equation to find the change in gravitational potential energy.

$\begin{array}{c}\Delta PE=\left(100\text{N}\right)\left(1\text{m}\right)\\ =100\text{J}\end{array}$

Thus, the change in the gravitational potential energy of the box for lifting the box straight up and pushing it up the ramp is $100\text{J}$ each.

Conclusion:

Thus the change in the gravitational potential energy of the box for lifting the box straight up and pushing it up the ramp is $100\text{J}$ each.

(g)

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

Work defines an activity involving mental or physical health but in physics it is defined as the force causing movement or displacement of an object. The work done by a given force is the product of the component of the force along the line of motion of the object multiplied by the distance that the object moves under the influence of the force.

When the box has to be lifted straight up, the person has to work against the complete gravitational attraction on the box or against the weight of the box. But when ramp is used work has to be done against only the component of the weight of the box along the ramp.

As obtained in the calculations, the force required to move the box to the final position is $100\text{N}$ when lifted straight up and $50\text{N}$ when the ramp was used. It is clear that the required force for obtaining the same result, in the case of ramp is only half of that in lifting straight up. This implies the person who is doing the work can conserve his strength by using the ramp.

Conclusion:

Thus 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.