Known

Unknown

First let us consider the force acting on the spring. Since the spring is hanging vertically, the only force acting on the spring is the force of gravity from the mass that has been added to the spring.

We can substitute in our variables and find the gravitational force.

According to Hooke’s Law the spring constant of a spring is directly proportional to the force acting on it and inversely proportional to the amount of stretch.

We can use the provided spring constant and the force acting on the spring to determine the amount of stretch.

Known

Unknown

First let us consider the force acting on the spring. Since the spring is hanging vertically, the only force acting on the spring is the force of gravity from the mass that has been added to the spring.

We can substitute in our variables and find the gravitational force.

According to Hooke’s Law the spring constant of a spring is directly proportional to the force acting on it and inversely proportional to the amount of stretch.

We can use the provided spring constant and the force acting on the spring to determine the amount of stretch.

First, we need to determine how fast the roller coaster must be going at the top of the loop to continue in a circular motion. At the top of the loop, the only force acting on the car is gravity. Therefore the gravitational force must be the cause of the centripetal motion.

We know that the force of gravity is

And the centripetal force equation is

We can set these two equations equal to each other.

Since mass is on both sides of the equation we can cancel it out.

We can rearrange and solve this equation for the velocity.

We can now use the conservation of energy to determine the initial height of the roller coaster. We know at the top of the roller coaster, there is only . At the top of the loop of the coaster there is both and .

Since mass is each factor, we can cancel it out.

The height of the coaster must start at .

First, we need to determine how fast the roller coaster must be going at the top of the loop to continue in a circular motion. At the top of the loop, the only force acting on the car is gravity. Therefore the gravitational force must be the cause of the centripetal motion.

We know that the force of gravity is

And the centripetal force equation is

We can set these two equations equal to each other.

Since mass is on both sides of the equation we can cancel it out.

We can rearrange and solve this equation for the velocity.

We can now use the conservation of energy to determine the initial height of the roller coaster. We know at the top of the roller coaster, there is only . At the top of the loop of the coaster there is both and .

Since mass is each factor, we can cancel it out.

The height of the coaster must start at .

For this we can consider the work-kinetic energy theorem. As work is done on the object, its kinetic energy is changing. In this case we have two different situations to consider. In the first we must consider the horizontal force acting on the box alone. In the second we must consider the horizontal force being resisted by a frictional force.

Let’s begin with the horizontal force acting alone.

Work is equal to the force times the displacement of the object.

In the first section the only force is and the displacement is .

We can use the work kinetic energy theorem to solve for the change in kinetic energy during this first section

Since the initial velocity is zero the equation becomes

We can now plug in our values

This is the velocity of the box after the first . Now it is time to analyze the motion of the box when it has both friction and the applied force.

Newton’s 2nd law says that the net force is equal to the sum of the forces involved.

We need to find the friction force.

The normal force in this case is equal to the force of gravity

We can now determine the work on the box through the next .

Like we did before we can now find the change of kinetic energy. This time we will use the final kinetic energy from the first part as the initial kinetic energy of the second part.

Therefore the box will have a final velocity of .

For this we can consider the work-kinetic energy theorem. As work is done on the object, its kinetic energy is changing. In this case we have two different situations to consider. In the first we must consider the horizontal force acting on the box alone. In the second we must consider the horizontal force being resisted by a frictional force.

Let’s begin with the horizontal force acting alone.

Work is equal to the force times the displacement of the object.

In the first section the only force is and the displacement is .

We can use the work kinetic energy theorem to solve for the change in kinetic energy during this first section

Since the initial velocity is zero the equation becomes

We can now plug in our values

This is the velocity of the box after the first . Now it is time to analyze the motion of the box when it has both friction and the applied force.

Newton’s 2nd law says that the net force is equal to the sum of the forces involved.

We need to find the friction force.

The normal force in this case is equal to the force of gravity

We can now determine the work on the box through the next .

Like we did before we can now find the change of kinetic energy. This time we will use the final kinetic energy from the first part as the initial kinetic energy of the second part.

Therefore the box will have a final velocity of .

We can use conservation of energy to solve this problem. Let us consider the types of energy at the beginning and the end. At the beginning the child has gravitational potential energy at the top of the slide. When the child reaches the bottom, the child has both kinetic energy and thermal energy as some energy was converted to heat because of the friction on the slide.

The law of conservation of energy states that we can set the energy at the beginning equal to the energy at the end.

This difference between the at the top and the at the bottom is the energy lost to friction.

We can use conservation of energy to solve this problem. Let us consider the types of energy at the beginning and the end. At the beginning the child has gravitational potential energy at the top of the slide. When the child reaches the bottom, the child has both kinetic energy and thermal energy as some energy was converted to heat because of the friction on the slide.

The law of conservation of energy states that we can set the energy at the beginning equal to the energy at the end.

This difference between the at the top and the at the bottom is the energy lost to friction.

To answer this question, we can address each type of energy separately. There is no conservation of energy in this problem; kinetic energy is not converted to potential energy as the man runs up the hill. Instead, he is accelerating, indicating an outside force that disallows conservation of energy.

First, we will find the maximum potential energy using the equation:

The man's mass and the acceleration of gravity will remain constant. The only changing variable is height. When the height is greatest, the potential energy will be the greatest. We can conclude that the potential energy will thus be greatest at the top of the hill.

Now we will look at the equation for kinetic energy:

The man's mass will remain constant, and the only changing variable will be the velocity. We are told that the man accelerates as he runs up the hill, indicating that his velocity is increasing. This tells us that he will reach a maximum velocity when he reaches the top of the hill, at which point he maintains a steady velocity along the top of the hill. Since kinetic energy is at a maximum when velocity is at a maximum, we can conclude that kinetic energy is greatest at the top of the hill.

To answer this question, we can address each type of energy separately. There is no conservation of energy in this problem; kinetic energy is not converted to potential energy as the man runs up the hill. Instead, he is accelerating, indicating an outside force that disallows conservation of energy.

First, we will find the maximum potential energy using the equation:

The man's mass and the acceleration of gravity will remain constant. The only changing variable is height. When the height is greatest, the potential energy will be the greatest. We can conclude that the potential energy will thus be greatest at the top of the hill.

Now we will look at the equation for kinetic energy:

The man's mass will remain constant, and the only changing variable will be the velocity. We are told that the man accelerates as he runs up the hill, indicating that his velocity is increasing. This tells us that he will reach a maximum velocity when he reaches the top of the hill, at which point he maintains a steady velocity along the top of the hill. Since kinetic energy is at a maximum when velocity is at a maximum, we can conclude that kinetic energy is greatest at the top of the hill.