Work and Energy - College Physics

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Question

A toy car is set up on a frictionless track containing a downward sloping ramp and a vertically oriented loop. Assume the ramp is tall. The car starts at the top of the ramp at rest.

What additional piece of information is necessary to calculate the maximum height of the loop if the car is to complete the loop and continue out the other side?

Answer

This is an example of conservation of energy. The car starts at the top of the ramp, at height . It has no velocity at this time since it is starting from a rest. Therefore its total energy is where is the mass of the car and is the value of gravitational acceleration.

At the bottom of the loop, all of the potential energy will have been converted into kinetic energy.

As the car traverses the loop and rises above the ground, kinetic energy will be converted back into potential energy. The shape of the loop does not matter in this case -- only the vertical distance between the ground and the car.

In the tallest possible loop, all kinetic energy at the bottom is converted to potential energy at the top. This is the maximum height the car can reach -- there is no additional energy left to continue climbing a taller loop. Therefore, the potential energy at the top of the tallest loop we can build is equal to the kinetic energy at the bottom of the loop. But we have already noted that the kinetic energy at the bottom of the loop is equal to the potential energy at the top of the ramp.

Therefore, we set . We see that and cancel, and we are left with . In other words, the tallest loop you can build is equal to the height of whatever ramp you select. In this example, the tallest loop we can build is . We do not need to know the specific values of or .

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Question

An elevator is designed to hold of cargo. The designers want the elevator to be able to go from the ground floor to the top of a tall building in . What is the minimum amount of power that must be delivered to the motor at the top of the shaft? Assume no friction and that the elevator itself has a negligible weight.

Answer

Power is the rate of energy transfer. To raise a object , a total of or ( is required. To find the power in Watts (), we divide the total energy required by the time over which the energy must be transferred:

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Question

How far can a person jump while running at and a vertical velocity of ?

Answer

We know that:

and we are looking for the maximum height (vertical displacement) this person can obtain, so we aren't concerned with .

We can apply the conservation of energy:

Masses cancel, so

Solve for :

(rounded to simplify our calculations)

so let's plug in what we know

. This is our final answer.

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Question

If a object has a kinetic energy of right after it is launched in the air, and it has KE at its max height, what is its max height?

Answer

Let's first write down the information we are given:

In order to solve this problem we must apply the conservation of energy, which states since no friction.

This means that as the project reaches its max height energy is converted from Kinetic Energy (energy of motion) to potential gravitational energy (based off of height).

We can subtract from to get the at its max height

=

so we can solve for the height

where

therefore

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Question

If an object has a kinetic energy of right after it is launched in the air, and it has KE at its max height of , what is the object's mass?

Answer

Let's first write down the information we are given:

In order to solve this problem we must apply the conservation of energy, which states since no friction.

This means that as the project reaches its max height energy is converted from Kinetic energy (energy of motion) to potential gravitational energy (based off of height).

We can subtract from to get the at its max height

=

so we can solve for the mass

where

therefore

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Question

A baseball weighing is dropped from a second story window which is high. What is the gravitational potential energy?

Answer

Gravitational potential energy is given by the equation:

We are given all the information needed to solve for the potential energy.

Plug in known values and solve.

Recall that the units for energy are Joules. Newtons is the unit for force.

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Question

A bowling ball is released from in the air. What is it's gravitation potential energy upon release?

Answer

The equation for gravitational potential energy is:

We are given all the information needed to answer the question.

Plug in known values and solve.

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Question

A bouncy ball is dropped from . When it bounces back up is reaches a height of . How much energy was loss?

Answer

The formula for gravitational potential energy is:

To find the potential energy lost, we need to find the potential energy of the ball at two heights, then find the difference.

Note that the energy was not actually lost; rather, it was converted to kinetic energy.

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Question

A car traveling at has a kinetic energy . If the car accelerates to , what will the new kinetic energy be?

Answer

Kinetic energy is given by. We will begin by calculating the car's initial kinetic energy, in terms of the unknown mass of the car :

.

Next, we will calculate the final energy of the car, also in terms of the unknown mass of the car:

.

To find the ratio of the final to initial kinetic energy, we divide by . We see that this reduces to with the both the mass and terms cancelling. . Thus, the new kinetic energy is .

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Question

A steel ball is in a spring loaded launcher. The spring has a constant of . If the ball is pulled back 50 cm then released and the ball leaves the launcher at a speed of , how much work was done by friction?

Answer

So in order to solve this problem, we first should think about where all the energy in the system is coming from and going. Specifically when you pull the spring back, the work you do is turned into potential energy stored in the spring. So:

and the energy transforms into kinetic when you let go, but some of that energy is lost to friction so

and

So use the equation below, enter in your known quantities and then solve for the work done by friction.

Plug in known values and solve.

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Question

Suppose that an object of mass is travelling towards a massless spring with a velocity . If the maximum displacement of the spring is , which of the following gives an expression for the spring constant, ?

Note: Assume that there is no friction.

Answer

For this question, we're told than an object is travelling with a certain velocity towards a spring. After colliding with the spring and causing it to undergo a maximum displacement, we're asked to provide an expression for the spring constant.

In order to answer this question, we'll need to consider the energy of the object and the spring during the process described. First, the object is moving with a certain velocity towards the spring. Thus, the object has kinetic energy. When the object collides with the spring and displaces the spring to its maximum displacement, there will be a brief instant where the object has completely stopped moving. At this point, all of the block's kinetic energy will have been transferred into the spring in the form of potential energy. With this information in mind, we can equate the two forms of energy.

Now, we can rearrange terms in order to isolate .

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