Chapter 5 Work and Energy

Chapter 5: Work and Energy 5.1 Work Done by a Constant Force 5.2 Work Done by a Variable Force 5.3 The Work-Energy Theorem: Kinetic Energy 5.4 Potential Energy 5.5 The Conservation of Energy 5.6 Power

5.1 Work Done by a Constant Force

5.1 Work Done by a Constant Force Definition: Work done by a constant force is equal to the product of the magnitudes of the displacement and the component of the force parallel to the displacement.

5.1 Work Done by a Constant Force

5.1 Work Done by a Constant Force W = (F cosӨ)d, where F is the magnitude of the force vector d is the magnitude of the displacement vector Ө is the angle between the two vectors (Warning – this angle is not necessarily measured from the horizontal) When the angle is zero, cos Ө = 1, and W = F·d When the angle is 180°, cos Ө = -1 and W = - F·d (yes, negative work!) example: the force of brakes to slow down a car. Although force and displacement are vectors, work is a scalar quantity. The SI unit of work is the N·m, which is called a joule (J).

5.1 Work Done by a Constant Force If there is no displacement, no work is done: W = 0. For a constant force in the same direction as the displacement, W = Fd. For a constant force at an angle to the displacement, W = (FcosӨ) d

5.1 Work Done by a Constant Force: Check for Understanding

5.1 Work Done by a Constant Force: Check for Understanding

5.1 Work Done by a Constant Force: Check for Understanding

5.1 Work Done by a Constant Force: Check for Understanding

Homework for Section 5.1 See Handout provide in class.

5.6 Power

Warmup: Rock-It

5.6 Work

5.6 Power A common British unit of power is the horsepower (hp) and 1 hp = 746 W.

5.6 Power

5.6 Power

5.6 Power Example 5.11: A 1500 kg car accelerates from 0 to 25 m/s in 7.0 s. What is the average power delivered to the car by the engine? Ignore all frictional and other losses.

5.6 Power: Check for Understanding

5.6 Power: Check for Understanding

5.6 Power: Check for Understanding

5.6 Power: Check for Understanding

Homework for Power See handout

The Work Energy Theorem: The Basic Model

Energy Energy is the capacity of a physical system to perform work. Energy exists in several forms such as heat, kinetic or mechanical energy, light, potential energy, electrical, or other forms. The SI unit of energy is the joule (J) or newton-meter (N * m). The joule is also the SI unit of work.

Law of Conservation of Energy According to the law of conservation of energy, the total energy of a system remains constant, though energy may transform into another form. Two billiard balls colliding, for example, may come to rest, with the resulting energy becoming sound and perhaps a bit of heat at the point of collision.

Basic Energy Model Within a system, energy can be transformed between various forms. Example #1: In a pendulum or swing (with the absence of frictional forces) Potential Energy  Kinetic Energy  Potential Energy Example #2: Potential energy of oil or gas is changed into energy to heat a building. In a closed system, the total energy in a system is constant and only transforms from on form of energy to another.

Basic Energy Model In an open system (most systems are open), energy can be transferred into and out of a system in 2 ways: Work: The transfer of energy by mechanical forces. Example: a golfer uses a club and gets a stationary golf ball moving when he or she hits the ball. The club does work on the golf ball as it strikes the ball. Energy leaves the club and enters the ball. This is a transfer of energy. Heat: The non-mechanical transfer of energy from a hotter object to a cooler object. Example: Brake on a car creating heat while stopping and the area around the brakes getting hot.

The Work Energy Theorem: Types of Mechanical Energy

K = kinetic energy (Joules – J) I. Kinetic Energy Kinetic Energy – Energy of motion KE = K = ½ mv2 K = kinetic energy (Joules – J) m = mass (kg) v = velocity (m/s) Note: KE cannot be negative, because… You cannot have a negative mass Velocity is always squared

I. Kinetic Energy

I. Kinetic Energy

II. Elastic Potential Energy Elastic potential energy is potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is equal to the work done to stretch the spring, which depends upon the spring constant k as well as the distance stretched.

II. Elastic Potential Energy PEe = Us = ½ kx2 PEe & Us = Elastic potential energy = Joules (J) k = spring constant (measures how stiff and strong the spring is) = N/m x = displacement of the spring from the rest or equilibrium position. (m) Note: PEe or Us cannot be negative because… k cannot have a negative value x is squared

II. Elastic Potential Energy

II. Elastic Potential Energy A child pulls back horizontally on a rubber band that has an unstretched length of 0.1 m and it stretches to a length of 0.25 m. If the spring constant of the rubber band is 10N/m, how much energy is stored in the spring?

III. Gravitational Potential Energy GPE is the energy associated with a gravitational field like the one we live in on Earth. AKA – Energy of position PEg = U = mgh PEg or U = gravitational potential energy – Joules (J) m = mass (kg) g = acceleration due to gravity (m/s2) h = height above ground (m) Note: GPE can be negative, but only if the object is below the horizontal line of the coordinate plane.

III. Gravitational Potential Energy

III. Gravitational Potential Energy A 10.0 kg object is moved from the second floor of a house 3.00 m above the ground to the first floor 0.30 m above the ground. What is the change in gravitational potential energy?

5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding

5.4 The Work-Energy Theorem: Potential Energy

5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding

5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding

5.4 The Work-Energy Theorem: Potential Energy: Check for Understanding Or, use opp = hyp (sin 30) = 0.5 m

Homework for Section Kinetic & Potential Energy Problems

B. Work Done by a Variable Force

Variable Forces: Spring Forces & Energy AP Physics I

I. Spring Forces Back and forth motion that is caused by a force that is directly proportional to the displacement. The displacement centers around an equilibrium position.

Springs – Hooke’s Law One of the simplest type of simple harmonic motion is called Hooke's Law. This is primarily in reference to SPRINGS. The negative sign only tells us that “F” is what is called a RESTORING FORCE, in that it works in the OPPOSITE direction of the displacement.

Example A load of 50 N attached to a spring hanging vertically stretches the spring 5.0 cm. The spring is now placed horizontally on a table and stretched 11.0 cm. What force is required to stretch the spring this amount? 110 N 1000 N/m

Hooke’s Law from a Graphical Point of View Suppose we had the following data: x(m) Force(N) 0.1 12 0.2 24 0.3 36 0.4 48 0.5 60 0.6 72 k =120 N/m

We have seen F vs. x Before!!!! Work or ENERGY = FDx Since WORK or ENERGY is the AREA, we must get some type of energy when we compress or elongate the spring. This energy is the AREA under the line! Area = ELASTIC POTENTIAL ENERGY Since we STORE energy when the spring is compressed and elongated it classifies itself as a “type” of POTENTIAL ENERGY, Us. In this case, it is called ELASTIC POTENTIAL ENERGY.

B. Work Done by a Variable Force The work done by an external force in stretching or compressing a spring (overcoming the spring force) is Work-Energy Thereom - WORK = D ENERGY (in a system) so, W = Us, which makes W = ½ kx2 where x is the stretch or compression distance and k is the spring constant. F F = kx work = area under the curve area of the triangle = ½ (base x height) work = ½ (x)(kx) = ½ kx2 slope = k x

Elastic Potential Energy The graph of F vs.x for a spring that is IDEAL in nature will always produce a line with a positive linear slope. Thus the area under the line will always be represented as a triangle. NOTE: Keep in mind that this can be applied to WORK or can be conserved with any other type of energy.

Conservation of Energy in Springs

Example A slingshot consists of a light leather cup, containing a stone, that is pulled back against 2 rubber bands. It takes a force of 30 N to stretch the bands 1.0 cm (a) What is the potential energy stored in the bands when a 50.0 g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does it leave the slingshot? 3000 N/m 60 J 49 m/s

B. Work Done by a Variable Force The reference position xo is chosen for convenience. xo may be chosen to be at the end of the spring in its unloaded position. b) xo may be at the equilibrium position when a mass is suspended on a spring. This is convenient when the mass oscillates up and down on the spring.

B. Work Done by a Variable Force Example: A spring of spring constant 20 N/m is to be compressed by 0.10 m. What is the maximum force required? What is the work required? Solution: Given: k = 20 N/m x = -0.10 m Unknown: a) Fs(max) b) W From Hooke’s law, the maximum force corresponds to the maximum compression. Fs(max) = -kx = -(20 N/m)(-0.10 m) = 2.0 N b) W = ½ kx2 = ½ (20 N/m)(-0.10 m)2 = 0.10 J

B. Work Done by a Variable Force: Check for Understanding 4 3 2 1

B. Work Done by a Variable Force

Homework for Work Done by a Variable Force See handout

5.5 The Conservation of Energy

Warmup: Road Hazard

5.5 The Conservation of Energy

5.5 The Conservation of Energy Remember, mechanical energy is kinetic or potential. If there is a nonconservative force such as friction doing work on the system, the total mechanical energy of the system is not conserved, but total overall energy is conserved.

5.5 The Conservation of Energy Example 5.8: A 70 kg skier starts from rest on the top of a 25 m high slope. What is the speed of the skier on reaching the bottom of the slope? (Neglect friction)

5.5 The Conservation of Energy If there is a nonconservative force doing work in a system, the total mechanical energy of the system is not conserved. However, the total energy (not mechanical!) of the system is still conserved. Some of the total energy is used to overcome the work done by the nonconservative force. The difference in mechanical energy is equal to the work done by the nonconservative force, that is Wnc = E – Eo = E

5.5 The Conservation of Energy Example 5.10: In Example 5.8, if the work done by the kinetic friction force is -6.0 x 103 J (the work done by kinetic friction force is negative because the angle between the friction force and the displacement is 180˚). What is the speed of the skier at the bottom of the slope?

5.5 The Conservation of Energy Example 5.9: A 1500 kg car moving at 25 m/s hits an initially uncompressed horizontal spring with spring constant 2.0 x 106 N/m. What is the maximum compression of the spring? (Neglect the mass of the spring.)

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

5.5 The Conservation of Energy: Check for Understanding

Explanation of the Math for Pulley Problem: 5.5 The Conservation of Energy Explanation of the Math for Pulley Problem: Uo = K Just block A: K = ½ m vA2 Both blocks together: K = ½ 3m vAB2 Since the change in energy is the same for both cases, set them equal to each other: ½ 3m vAB2 = ½ m vA2 vAB2 = vA2/ 3 vAB = vA / √3

Homework for Section 5.5 See handout

Chapter 5 Formulas K = ½ mv2 kinetic energy Ug = mgh gravitational potential energy W = F r cos  work Pave = W power as defined by work over time t P = Fv cos  power if the force is making an angle  with v Fs = -kx Hooke’s Law; relates spring force with spring constant and change in length Us = ½ kx2 elastic potential energy