Mechanics - Physics
Card 0 of 228
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?
A toy car is set up on a frictionless track containing a downward sloping ramp and a vertically oriented loop. Assume the ramp is
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?
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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 
.
This is an example of conservation of energy. The car starts at the top of the ramp, at height
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
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.
An elevator is designed to hold
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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:
Power is the rate of energy transfer. To raise a
How far can a person jump while running at 
and a vertical velocity of 
?
How far can a person jump while running at
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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.
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
so let's plug in what we know
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?
If a
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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
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
so we can solve for the height
where
therefore
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?
If an object has a kinetic energy of
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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 
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
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
so we can solve for the mass
where
therefore
A baseball weighing 
is dropped from a second story window which is 
high. What is the gravitational potential energy?
A baseball weighing
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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.
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.
A 
bowling ball is released from 
in the air. What is it's gravitation potential energy upon release?
A
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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 equation for gravitational potential energy is:
We are given all the information needed to answer the question.
Plug in known values and solve.
A pendulum on earth has a period of 
. What is it's period on Mars with it's gravity is 
A pendulum on earth has a period of
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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:
First, we need to find the length of the pendulum. Begin with the equation for finding the period of a pendulum:
solve for
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:
A 
bouncy ball is dropped from 
. When it bounces back up is reaches a height of 
. How much energy was loss?
A
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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.
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.
A car traveling at 
has a kinetic energy 
. If the car accelerates to 
, what will the new kinetic energy be?
A car traveling at
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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 
.
Kinetic energy is given by
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
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?
A
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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.
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
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.
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.
Suppose that an object of mass
Note: Assume that there is no friction.
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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 
.
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
A pendulum is made up of a small 
mass that hangs on the end of a 
long string of negligible mass. The pendulum is displaced by 
and allowed to undergo harmonic motion. What is the angular frequency of the resulting motion?
A pendulum is made up of a small
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The angular frequency of a simple pendulum is 
, where 
is the length of the pendulum.
The angular frequency of a simple pendulum is
For a simple harmonic motion governed by Hooke's Law, 
, if 
is the period then the quantity 
is equivalent to which of the following?
For a simple harmonic motion governed by Hooke's Law,
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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 T is the period. The equation for T is
Solve for
If the mass of a simple pendulum is quadrupled, then its period .
If the mass of a simple pendulum is quadrupled, then its period .
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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.
We know that the equation for the period of a simple pendulum is
A violin string 
long has a linear density of 
. What is the string tension if the second harmonic has a frequency of 
?
A violin string
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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 
.
Since we are solving for string tension, we need to use the frequency equation with the tension variable in it. That equation is
We are given:
Next we must convert the length of
An hourglass is placed on a scale with all its sand in the upper chamber. A short time later, the sand begins to fall into the lower chamber. Which of the following best describes the reading on the scale as a function of time before any sand has accumulated in the bottom chamber?
An hourglass is placed on a scale with all its sand in the upper chamber. A short time later, the sand begins to fall into the lower chamber. Which of the following best describes the reading on the scale as a function of time before any sand has accumulated in the bottom chamber?
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Initially, when all the sand is in the upper chamber, the reading on the scale is constant and corresponds to the weight of the hourglass and the sand within. As some sand falls, there is no normal force on it so the scale can not register its weight. The fraction of sand that is falling is small, so there is only a small decrease in the apparent weight of the hourglass.
Initially, when all the sand is in the upper chamber, the reading on the scale is constant and corresponds to the weight of the hourglass and the sand within. As some sand falls, there is no normal force on it so the scale can not register its weight. The fraction of sand that is falling is small, so there is only a small decrease in the apparent weight of the hourglass.
Which of the following is not an example of simple harmonic motion?
Which of the following is not an example of simple harmonic motion?
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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.
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.
The bases on a baseball field are 
apart in a perfect square.
A player hits a home run and goes around all 4 bases. What is the total distance he travelled?
The bases on a baseball field are
A player hits a home run and goes around all 4 bases. What is the total distance he travelled?
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There is 90 feet between each base and distance does not depend on direction. Distance is scaler therefore, the batter travels 360 feet.
There is 90 feet between each base and distance does not depend on direction. Distance is scaler therefore, the batter travels 360 feet.
The bases on a baseball field are 
apart.
A player hits a home run and runs around all four bases. What is his total displacement?
The bases on a baseball field are
A player hits a home run and runs around all four bases. What is his total displacement?
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Displacement is a vector. Therefore, magnitude and direction matters and because direction matters the total displacement is 0 feet.
Displacement is a vector. Therefore, magnitude and direction matters and because direction matters the total displacement is 0 feet.