Physics

Conservative and

Non-Conservative

Forces

Teacher: Luiz Izola

Chapter Preview

1. Conservative Forces

2. Non-Conservative Forces

3. Potential Energy

4. Work Done by Conservative Forces

5. Conservation of Mechanical Energy

6. Work Done by Non-Conservative Forces

7. Potential Energy Curves and Equipotentials

Introduction

One of the greatest physics concepts is the conservation of energy.

Energy has several forms (mechanical, thermal). The universe has a constant amount of energy that flows from one form to another.

In this chapter we will focus on the conservation of energy, the first “conservation law”.

Conservative Forces The work done by a conservative force can be stored in the form of energy and released at a later time. Simplest case of conservative force Gravity

Conservative Forces

Definition1: A conservative force does zero work on a closed path.

Definition2: Work done by a conservative force in going from points A to B is independent of the path taken.

Conservative Forces

W1 + W2 = 0

W1 + W3 = 0

Therefore: W2 = W3

Non-Conservative Forces The work done by a non-conservative force cannot be recovered later as kinetic energy.

It is converted to other forms of energy such as heat. See the work done by friction below.

Non-Conservative Forces Considering the same closed path but analyzing

the effect of friction on the total work, we would have: Wtotal = -4μkmgd

Example

A 4.57kg box is moved with constant speed from A to B along the two paths below. Calculate the work done by gravity on each one of these paths. Also, calculate the work done by friction (μk = 0.63)

along the two paths.

Potential Energy (U) Potential energy is a storage system for energy.

Energy is never lost as long as the separation remains the same. For example, when we lift a ball we produce an amount of potential energy related to the height we lift it. The ball can rest on a shelf for a million years and when it falls, it will gain the same amount of kinetic energy.

Work done against friction is not stored as potential energy. It dissipates as heat or sound.

Only conservative forces have the potential-energy storage system.

Work by Conservative Forces When a conservative force does an amount of work Wc (c = conservative), the corresponding potential energy U is changed according to:

Wc = Ui – Uf = -ΔU

The work done by a conservative force is equal to

the negative of the change in potential energy.

For example, when an object falls, gravity does positive work on it and its potential energy decreases.

Work by Conservative Forces

Gravity: Let’s apply our definition of potential energy to the force of gravity near the Earth’s surface.

Work by Conservative Forces Gravitational Potential Energy

U = mgy

y = height

m = mass

g = acceleration of gravity

Example: Find the gravitational potential energy of a 65-kg person on a 3.0meter diving board. Let U = 0 be the water level.

Work by Conservative Forces

Example: An 82kg mountain climber is in the final stage of a 4301-meter-high peak. What is the change in potential energy as the climber gains the last 100 meters of altitude? Let U = 0 be (a) at sea level (b) at the top of the peak.

Work by Conservative Forces Springs: Consider a spring that is stretched from

its equilibrium position a distance x. The work required to cause this stretch is W = 1/2Kx2.

From our definition of potential energy, we have:

Wc = 1/2Kx2 = Ui – Uf

Assuming that at x=0 (equlibirum position), U = 0, we can simplify the formula: Ui = 1/2Kx2 .

Spring Potential Energy: U = 1/2Kx2

Work by Conservative ForcesExample: Find the potential energy of a spring with force constant k=680N/m if it is (a) stretched 5cm and (b) compressed 7cm.

Example: When a force of 12.0N is applied to a certain spring, it causes a stretch of 2.25cm. What is the potential energy of this spring when it is compressed by 3.50cm?

Practice Session

1) Calculate the work done by the gravity as a 2.6kg object is moved from point A to point B, along paths 1,2, and 3.

Practice Session

2) Calculate the work done by friction as a 2.6kg box is slid along a floor from point A to point B along paths 1,2, and 3. Assume the kinetic coefficient of friction is 0.23. Based on previous picture.

3) A 4.1kg block is attached to a spring with force constant of 550N/m. Find the work done by the spring on the block as the block moves from A to B along paths 1 and 2.

Practice Session

4) Calculate the work done by gravity as a 5.2kg object is moved from A to B along paths 1 and 2. How does the mass of the object affects the results?

Practice Session

5) As an Acapulco cliff diver drops to the water from a height of 40.0meters, his gravitational potential energy decreases by 25,000J. How much does the diver weighs?

6) Find the gravitational potential energy of an 80.0kg person standing atop Mt. Everest, at an altitude of 8848meters. Use the sea level as the location y = 0.

7) Compressing a spring 0.50cm produces a potential energy equals 0.0035J. Which compression is required to generate a energy equals 0.080J?

Practice Session

8) A force of 4.7N is required to stretch a certain spring by 1.30cm. (a) How far must the spring be stretched for its potential energy to be 0.020J? (b) How much stretch is required for the spring potential energy to be 0.080J?

9) A 0.33kg pendulum bob is attached to a string 1.2meters long. What is the bob’s potential energy change from points A to B?

Conservation of Mechanical Energy Mechanical Energy is the sum of potential and

kinetic energies of an object.

E = U + K Mechanical energy is conserved on systems

involving ONLY conservative forces. E is constant in this case.

Proving that E is constant for conservative forces

Wtotal = ΔK = Kf – Ki and Wtotal = Wc

We know that: Wc = Ui – Uf . Then, replacing both,

we get: Kf – Ki = Ui – Uf .

Rearranging, we get: Ef = Ei.

Conservation of Mechanical Energy

In terms of physical systems, conservation of mechanical energy means that energy can be converted between potential and kinetic forms, but the SUM REMAINS THE SAME.

In systems with conservative forces only, the mechanical energy E is conserved, that is:

E = U + K = constant

Example

ExampleEi = Ef

Ui + Ki = Uf + Kf

From left side of previous picture, we have:

mgh + 0 = 0 + 1/2mv2

Therefore: v = (2gh)1/2

Now: Prove for right side of previous picture.

Practice

At the end of a graduation ceremony, graduates fling their caps into the air. Suppose a 0.12kg cap is thrown straight upward with an initial speed of 7.85m/s and there is no friction. (a) Use kinematics to find the speed of the cap when is 1.18m above the release point. (b) Show that mechanical energy is the same at the release and at 1.18m

PracticeIn the bottom of the ninth inning, a player hits a 0.15kg

baseball over the outfield fence. The ball leaves the bat with a speed of 36m/s, and a fan in the bleachers catches it 7.2m above the point where it was hit. Find (a) Kinetic Energy when the ball is caught. (b) The speed when it is caught.

PracticeA 55kg skateboarder enters a ramp

horizontally with a speed of 6.5m/s, and leaves the ramp vertically with a speed of 4.1m/s. Find the height of the ramp.

PracticeA 1.7kg block slides on a horizontal, frictionless

surface until it encounters a spring with a force constant k=955n/m. The block comes to rest after compressing the spring a distance of 4.60m. Find the initial speed of the block.

Practice

Suppose the spring and block are oriented vertically. Initially, the spring is compressed 4.60cm and the block is at rest. When the block is released, it accelerates upward. Find the speed of the block when the spring returns to the equilibrium position.

Homework1. If a 30.0J of work is required to stretch a spring

from 4.00cm to 5.00cm., how much work is necessary to stretch it from 5.00cm to 6.00cm?

2. A spring scale has a spring with a force constant of 250N/M and a weighing pan with a mass of 0.075kg. During the first weighing, the spring is stretched a distance of 12cm from equilibrium. The second time is stretched 18cm. (a) How much greater is the elastic potential energy of the spring during the second than the first weighing? (b) If the spring is released after each weighing. What is the ratio of the pan’s maximum speed between second and first weighing?

Homework3. An 80.0N box of clothes is pulled 20.0m up a

30o ramp by a force of 115N that points along the ramp. If the coefficient of kinetic friction between the box and the ramp is 0.22, calculate the change in the boy’s kinetic energy.

4. A 0.60kg rubber ball has speed of 2.0m/s at point A and kinetic energy of 7.5J at point B. Determine the following: (a) Ball’s kinetic energy at A. (b) Ball’s speed at B. (c) Work done by ball from A to B.

Homework5. Starting from rest, a 5.0kg block slides 2.5m

down a rough 30o incline in 2.0s. Determine the following: (a) The work done by gravity. (b) The mechanical energy lost due to friction

6. At a park, a swimmer uses a water slide to enter the main pool. If the swimmer starts at rest, slides without friction, and falls through a vertical height of 2.61m, what is her speed at the bottom of the slide?

7. A player passes a 0.60kg ball. The ball leaves the player’s hands with a 8.30m/s speed and slows to 7.10m/s at its highest point. How high is the ball at the highest point from release?

Homework8. An 18kg child plays on a slide that drops

through a height of 2.2m. The child starts at rest. On the way down a non-conservative work of -373J is done on the child. What is the child’s speed at the bottom of the slide?

9. A 17,000kg airplane lands with a speed of 82m/s on a 115m long carried deck. Find the work done by non-conservative forces in stopping the plane.

10. A 5.0kg rock is dropped and allowed to fall freely. Find the initial kinetic energy, the final kinetic energy, and the change in kinetic energy for (a) first 2 meters of the fall. (b) next two meters of the fall