Gravitational Potential Energy - AP Physics 1

This problem can either be very simple or very complex.

You could convert the contractor's potential energy into kinetic energy, and then find the velocity at which he hits the seesaw. Then, use that to calculate the kinetic energy of the bricks and the height at which the bricks will fly. Although some may find that fun to do, it's unnecessary.

If we assume that all of the energy of the man is transferred to the bricks, we can use the conservation of energy equation:

We need to make a few clarifying statements for our initial and final states. In the initial state, the contractor is at a height of 1.5m and not moving. The bricks are not moving at this point either. In the final state, the man is on the ground not moving, and the bricks are at a height of 10m and not moving. Therefore, we can rewrite the above equation as the following:

Rearranging for the mass of the bricks:

Using the equation for conservation of energy, we get:

Let's clarify that our initital state is when the ball is at its maximum height and the final state is when it reaches the ground. Plugging in our expressions and removing initial kinetic and final potential energy, we get:

Note that we removed initial kinetic energy despite the ball moving. At the initial state, all of the ball's velocity is horizontal. Since there is no vertical velocity, we can ignore it for now, coming back to it later.

We multiply the initial energy by 0.85 because the problem statement says that we lose 15% of the ball's total energy after hitting the rim. Rearranging for final velocity:

This is only the y-component of the final velocity. We need to combine this with the x-component to get the total velocity. The problem statement tells us that the final horizontal velocity is , therefore we can write:

This problem helps you become more comfortable with using obscure units. Since we are given work as a function of mass, we don't actually have to know the mass of the block. Furthermore, since we are neglecting air resistance, we know that all of the work is done by friction. To calculate the force of friction, we can use the following expression:

To calculate the work done by this force, we multiply by the distance that it was applied:

Since we were given work as a function of mass, we can eliminate mass to get:

Rearranging for the angle, we get:

We know all of these values, allowing us to solve for the angle:

We can use the expression for conservation of energy to solve this problem:

Assuming a final height of zero, we can eliminate final potential energy. Then, substituting in expressions for each variable, we get:

Canceling out mass and rearranging for final velocity, we get:

The change in gravitational potential energy is only dependent on the final and initial height of the object, and is independent of the path taken to get to the final height. Because the change in height for the rock is zero, the change in gravitational potential energy equals zero.

Over the distance described, change in gravitational pull is negligible, so gravity can be treated as a constant. Potential energy therefore varies with the dumbell's proximity to the ground, h:

Since the dumbell has an inital position of 100m, and an initial velocity of zero, its height can be described using the kinematic equation:

From this, the potential energy at a time of three seconds is:

Gravitational potential energy is proportional to both the height and mass of an object. Gravitational potential energy is given by:

Gravitational potential energy is also relative to a "zero height" and in our case this is the ground. What this means is that if the ground is a our "zero height", the higher the object from the ground the greater the potential energy

Gravitational potential energy is proportional to both the height from a "zero potential energy" reference point (in our case the floor) and the mass. While the balls would have the same gravitational potential energy since they are the same mass, they have differing energies because of differing heights. The shortest cord, or the ball the highest from the ground will have the greatest potential energy as shown:

The first step for this problem is to determine the potential energy the ball has at the top of the ramp through this equation:

We see that it has a potential energy of 100 joules. All of this potential energy gets converted to kinetic energy as the ball falls down the ramp.

Knowing this, we can determine the velocity the ball has at the bottom of the ramp by setting the potential energy equal to the kinetic energy:

We substitute the known mass and solve for :

Use conservation of energy:

Plug in known values and solve for the maximum height of the object.

Recall that the formula for gravitational potential energy is:

To determine the height on planet Mars needed to have equivalent potential energy, we can set the two equations for gravitational potential energies equal.

, where is mass and is height above the ground on Earth while is the height above the ground on Mars.

The mass, can be eliminated from both sides of the equation since they are equal. Plug in and solve for the height on Mars.

Use conservation of energy:

Since initially the cat is on the ground, and can assumed to be still at the top of the wall:

Solve for velocity:

Plugging in values

Assume a constant acceleration we may use a kinematic formula:

is the acceleration due to gravity, which is

, the initial velocity, is zero

is the unknown

is the ground, which will be a height of

Plug in values:

Solve for

Gravitational potential energy is calculated using the following equation.

Where is the potential energy, is the gravitational constant, is the mass of one object, is the mass of the other object, and is the distance between the two objects. Decreasing and increasing mass of the object by the same factor will have a similar effect on the potential energy (both will increase potential energy by the same factor); therefore, changing both of these variables by same factor will have a similar effect. Recall that the force due to gravity is

When calculating , will have the biggest effect. Don't confuse equation for potential energy with equation for force due to gravity.

Potential energy is calculated using the following equation.

Where is the potential energy, is the gravitational constant, is the mass of one object, is the mass of the other object, and is the distance between the two objects. The potential energy depends on the mass of the objects (in this case the person and the planet); therefore, the more massive planet will produce the higher potential energy.

The mass is the same regardless of the situation. Mass measures the amount of substance/elements/molecules inside a person or object's body. The composition of the person’s body won’t change (regardless of the mass of the planet); therefore, the mass will stay the same. Recall that the weight of the person, however, does change based on the planet the person is standing on. This is because the weight depends on the gravitational acceleration, which depends on the mass of the planet.

The potential energy of a person or an object on earth is calculated using the following formula.

Where is potential energy, is mass, is acceleration due to gravity, and is the height from surface of earth (sea level). The question states that the person is standing on the surface of earth; therefore, . This means that the person has no potential energy.

Recall that kinetic energy, on the other hand, is the energy associated with motion. It depends on the velocity of an object or person. A person on the surface of earth will have kinetic energy if the person is moving.

Definition of work:

Combine equations:

Substitute variables, then plug in and solve for the change in heat.

At the top of the loop, the centripetal acceleration will need to be at least the acceleration due to gravity:

Solve for

The radius of the circle will be half the height:

Use conservation of energy:

Solve for :

Combine equations:

Plug in values:

Using conservation of energy:

Solving for :

Plugging in values: