Mechanics - Physics

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 .

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:

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.

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

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

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.

The equation for gravitational potential energy is:

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

Plug in known values and solve.

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.

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 .

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.

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 .

The angular frequency of a simple pendulum is , where is the length of the pendulum.

We know that T is the period. The equation for T is for harmonic motion.

Solve for by dividing the equation by on both sides. The result is , which is the answer.

We know that the equation for the period of a simple pendulum is . This equation does not depend on mass. It is only affected by the length of the pendulum (L) and the gravitational constant (g). Therefore, adding mass to the pendulum will not effect the period, so the period remains the same.

Since we are solving for string tension, we need to use the frequency equation with the tension variable in it. That equation is where is the frequency, is the number of the harmonic, is the length of the string, is the linear density of the string, and is the tension of the string.

We are given:

Next we must convert the length of to meters which is and the mass density of to . Then we plug in our known values into the equation and solve for the string tension. The result is .

First, we need to find the length of the pendulum. Begin with the equation for finding the period of a pendulum:

solve for to get:

Now we can plug in our given values:

Since we have the length of the pendulum determined, we can now find the period of the pendulum on Mars:

For this question, we need to recall what simple harmonic motion is. Remember that it is a periodic motion where the restoring force depends on the displacement of the object undergoing these motions. So to answer this question, we need to keep this idea in mind and see which example doesn't match up.

A mass on a pendulum moving back and forth is clearly an example of simple harmonic motion. As the mass moves further from the center in either direction, it experiences a greater and greater force in the opposite direction.

A child swinging on a swing set is another correct example. This situation is analogous to the mass on a pendulum swinging back and forth.

A vibrating guitar string is yet another example of simple harmonic motion. After it is plucked, the string oscillates back and forth.

Finally, a book falling to the ground does not represent harmonic motion. Once the book is released from rest, we intuitively know that it will fall to the ground and will then stay there; in no way is there any periodic motion.

There is 90 feet between each base and distance does not depend on direction. Distance is scaler therefore, the batter travels 360 feet.

Displacement is a vector. Therefore, magnitude and direction matters and because direction matters the total displacement is 0 feet.