Chapter 6

Potential Energy and Conservative Forces

Conservative Force: Definition 1
A conservative force does zero total work on any closed path.

Conservative Force Definition 2
The work done by a conservative force in going from an arbitrary point A to an arbitrary point B is independent of the path from A to B.

Potential Energy
Potential Energy is energy stored through the interaction of two or more material objects. Potential Energy is retrievable stored energy, energy by virtue of position or configuration in relation to a force.

The change in the potential energy of a body incurred in moving from one point to another equals the work done in overcoming the interaction (gravitational, electromagnetic, strong, or weak) that stores the energy.

Gravitational Potential Energy ()

This equation works if the object is close to the Earth, but g is not constant. Therefore, to get a more accurate value we must use Newton's Universal Law Gravitation.

For most of our calculuations the first equation is sufficient, but if the object is to high or the motion has a large displacement to a point that will have a different g then the normal g the second equation will be used.

Example 15

Find the gravitational potential energy of a 65 kg person on a 3.0 m high diving board. Let U = 0 be at water level.

Example 16

An 82 kg mountain climber is in the final stage of the ascent of 4301 m high Pikes Peak. What is the change in gravitational potential energy as the climber gains the last 100 m of altitude? Let = 0 be (a) at sea level or (b) at the top of the peak.

Example 17

A candy bar called the Mountain Bar has a calorie content of 210.0 Cal = 210.0 kcal, which is equivalent to an energy of . If an 82 kg mountain climber eats a Mountain Bar and magically converts it all to potential energy, what gain of altitude would be possible?

Example 18

The figure shows a 2.0 kg block of slippery cheese that slides along a frictionless track from point a to point b. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m. How much work is done on the cheese by the gravitational force during the slide?

Example 19

A 2.0 kg sloth hangs 5.0 m above the ground. See figure.
(a) What is the gravitational potential energy U of the sloth-Earth system if we take the reference point y = 0 to be (1) at the ground, (2) at a balcony floor that is 3.0 m above the ground, (3) at the limb, and (4) 1.0 m above the limb? Take the gravitational potential energy to be zero at y = 0.
(b) The sloth drops to the ground. For each choice of reference point, what is the change in the potential energy of the sloth-Earth system due to the fall?

Conservation of Mechanical Energy

Conservation of Energy:
The total energy of any system that is isolated from the rest of the Universe remains constant, even though energy may go from one form to another within the system.

You are not limited to gravitational potential energy. There is electrical potential, thermal energy etc.

If friction is involved in the motion we need to add that into the thought process for solving the problem.

Using Newton's Second Law and friction we have:

Since, the force and acceleration are constant we can use one of the kinematic equations and solve for the acceleration.

Then substitute this into Newton's Second Law equation and performing some algebra.

The first two terms are the final and initial kinetic energy. Also we could include potential energies so:

By some experimentation we would see that the amount of thermal energy in the system from moving the object is equal to the work done by friction. So we can replace the frictional work with the change in thermal energy.

Example 20

In the figure a child of mass m is released from rest at the top of a water slide, at height h = 8.5 m above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.

Example 21

A food shipper pushes a wood crate of cabbage heads (total mass m = 14 kg) across a concrete floor with a constant horizontal force of magnitude 40 N. In a striaght-line displacement of magnitude d = 0.50 m, the speed of the crate decreases from to . (a) How much work is done by force ? (b) What is the increase in the thermal energy of the crate and floor?

Example 22

At the end of a graduation ceremony, graduates fling their caps into the air. Suppose a 0.120 kg cap is thrown straight upward with an initial speed of 7.85 m/s, and that frictional force can be ignored. (a) Use kinematics to find the speed of the cap when it is 1.18 m above the release point. (b) Show that the mechanical energy at the release point is the same as the mechanical energy 1.18 m above the release point.

Example 23

In the bottom of the ninth inning, a player hits a 0.15 kg baseball over the outfield fence. The ball leaves the bat with a speed of 36 m/s, and a fan in the bleachers catches it 7.2 m above the where it was hit. Assuming frictional forces can be ignored, find (a) the kinetic energy of the ball when it is caught and (b) its speed when caught.

Example 24

Swimmers at a water park can enter a pool using one of two frictionless slides of equal height. Slide 1 approaches the water with a uniform slope; slide 2 dips rapidly at first, then levels out. Is the speed at the bottom of slide 2 (a) greater than, (b) less than, or (c) the same as the speed at the bottom of slide 1?

Example 25

A 55 kg skateboarder enters a ramp moving horizontally with a speed of 6.5 m/s, and leaves the ramp moving vertically with a speed of 4.1 m/s. Find the height of the ramp, assuming no energy loss to frictional forces.

Example 26

A snowboarder coasts on a smooth track that rises from one level to another. If the snowboarder's initial speed is 4 m/s, the snowboarder just makes it to the upper level and comes to rest. With a slightly greater initial speed of 5 m/s, the snowboarder is still moving to the right on the upper level. Is the snowboarder's final speed in this case (a) 1 m/s, (b) 2 m/s, or (c) 3 m/s?

Example 27

A block of mass is connected to a second block of mass , as shown in the figure. When the blocks are released from rest they move through a distance d = 0.500 m, at which point hits the floor. Given that the coefficient of kinetic between and the horizontal surface is , find the speed of the blocks just before lands.

Power

Power is the rate of doing work.

The units of power are a joule/second which we define as a watt (W).

1 hp = 550 ft•lb/s = 746 W = 0.746 kW

When a constant force is applied to an object we can use the average velocity and the force to find the power.

Example 28

To pass a slow-moving truck, you want your fancy, car to accelerate from 13.4 m/s to 17.9 m/s in 3.00 s. What is the minimum power required for this pass?

Example 29

It takes a force of 1280 N to keep a 1500 kg car moving with a constant speed up a slope of . If the engine delivers 50.0 hp to the drive wheels, what is the maximum speed of the car?

Example 30

The figure shows constant forces and acting on a box as the box slides rightward across a frictionless floor. Force is horizontal, with magnitude 2.0 N; force is angled upward by to the floor and has magnitude 4.0 N. The speed v of the box at a certain instant is 3.0 m/s. (a) What is the power due to each force acting on the box at that instant, and what is the net power? Is the net power changing at that instant? (b) If the magnitude of is, instead, 6.0 N, what now is the net power, and is it changing?

Example 31

In the figure, a circus beagle of mass m = 6.0 kg runs onto the left end of a curved ramp with speed at the initial height of 8.5 m above the floor. It then slides to the right and comes to a momentary stop when it reaches a height y = 11.1 m from the floor. The ramp is not frictionless. What is the increase in the thermal energy of the beagle and ramp because of the sliding?

The End