Work and Energy Chapter 7

Conservation of Energy Energy is a quantity that can be converted from one form to another but cannot be created or destroyed. The total amount of energy present in any process remains the same.

Work Has a very specific definition in physics. Involves a force moving a body through a displacement. If the force is in the same direction as the motion,

Vector equation for work If the force is directed at an angle to the displacement, then only the component of the force that is parallel to the displacement does work. F

Vector equation for work Important – work is a scalar quantity, not a vector, even though it is derived from two vectors. Work does not have a direction.

Work If  is between 0° and 90°, cos  is positive, so work done is positive. If  is between 90° and 180°, cos  is negative, so work done is negative. If  is 90°, cos  is zero, so work done is zero.

Example Lifting, carrying, and lowering a book.

Units on work The unit for work is a joule.

Example During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled 5 m. Find the amount of work you do

On your own A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m. Find the work done by the crane on the truck

Work and energy with varying forces The equations we’ve developed so far for work involve constant forces and motion along a straight line. Now let’s look at varying forces. We could also look at motion along curved lines, but we would need to use integrals, which you don’t know how to do yet.

The spring force The farther you stretch a spring, the harder it is to stretch. The force you must apply follows the following equation, known as Hooke’s Law Where k is called the force constant of the spring or the spring constant. k has units of N/m

Work done on a spring Where X is the total elongation of the spring. See page 151.

Work done on a spring Where x 1 is the initial position of the spring, x 2 is the final position of the spring x = 0 is the equilibrium position of the spring (when it is neither stretched nor compressed.)

Example A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x 1 = -5 cm. Find the work done on the block as the block moves to its equilibrium position x 2 = 0

On your own To stretch a spring 3.00 cm from its unstretched length, 12.0 J of work must be done. How much work must be done to compress this spring 4.00 cm from its unstretched length?

Work and Kinetic Energy The total work done on a body is related to changes in the speed of the body.

Work and Kinetic Energy

Work-Kinetic Energy Theorem The quantity ½ mv 2 is called the kinetic energy and represented by the letter K. It is a scalar quantity and can never be negative. It is zero when an object is not moving.

Units on kinetic energy Looking at the equation, we have

Work-kinetic energy theorem The work-kinetic energy theorem still works for varying forces. See the last paragraph on page 153 for a mathematical explanation of this.

Motion along a curved path Where dl is a distance along the path The work-kinetic energy theorem still holds true.

Example During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled for 5 m. Find The work you do The final speed of the sled

On your own A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m. Find The work done by the crane on the truck The total work done on the truck The upward speed of the truck after the 2 m if it started from rest

Example A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x 1 = -5 cm. Find The work done on the block as the block moves to its equilibrium position x 2 = 0 The speed of the block at x 2 =0

Where does kinetic energy come from? If an object falls off a cliff, gravity does work on it as it falls, and increases its kinetic energy. But where does that energy come from?

Potential Energy Energy seems to be stored in some form related to height. This energy is related to the position of a body, not its motion. It is called potential energy. – measures potential or possibility for work to be done. (Some kinds of potential energy are related to things other than height.)

Work done by gravity Work done by gravity as a body falls from height y 1 to height y 2. Also works if object is rising – then y 2 is greater than y 1, so the work is negative, as it should be.

Gravitational Potential Energy We define the gravitational potential energy as So, the work done by gravity is

If only gravity does work

Conservation of Mechanical Energy When only gravity does work, the total mechanical energy is constant.

Where do I measure y from? Wherever you want. It is only the change in potential energy that we are interested in, not the value of U at a particular point. So choose your zero potential energy point wherever it is convenient.

Example Standing near the edge of the roof of a 12-m high building, you kick a ball with an initial speed of v i = 16 m/s at an angle of 60° above the horizontal. Neglecting air resistance, use conservation of energy to find How high above the height of the building the ball rises Its speed just before it hits the ground 9.79 m22.2 m/s

You try You throw a kg ball straight up in the air, giving it an initial upward velocity of 20.0 m/s. Use conservation of energy to find how high it goes m

Effect of other forces If forces other than gravity are acting on a body

Example A child of mass 40 kg goes down an 8.0 m long slide inclined at 30° above the horizontal. The coefficient of kinetic friction between the slide and the child is If the child starts from rest at the top of the slide, how fast is she traveling when she reaches the bottom? 5.60 m/s

Curved path The shape of the path makes no difference for gravitational potential potential energy. You can still use the same equations for conservation of energy.

On your own A pendulum consists of a bob of mass m attached to a string of length L. The bob is pulled aside so that the string make an angle  0 with the vertical, and is released from rest. Find an expression for its speed as it passes through the bottom of the arc. 00

Elastic potential energy Think slingshot. The farther you stretch it, the more kinetic energy the projectile can gain.

Springs For springs, the elastic potential energy is given by The work done by a spring is The equations work whether the spring is stretched or compressed.

Where is x = 0? We don’t get to choose this time x = 0 is always at the equilibrium position of the spring – when it is neither stretched nor compressed.

Conservation of energy The conservation of energy works the same for springs as it does for gravity.

Conservation of energy If we have both gravitational and elastic potential energies, then

Example A 1.20-kg piece of cheese is placed on a vertical spring of negligible mass and force constant k = 1800 N/m that is compressed 15.0 cm. When the spring is released, how high does the cheese rise from its initial position? (The spring and the cheese are not attached.) 1.72 m

On your own A 2000-kg elevator with broken cables is falling at 25 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 3.00 m as it does so. During the motion, a safety clamp applies a constant 17,000 N frictional force to the elevator. What is the force constant of the spring? 1.41 x 10 5 N/m

Conservative forces The work done by conservative forces: Can always be expressed as the difference between the initial and final values of a potential energy function. Is reversible Is path-independent (It only depends on the starting and ending points.) Equals zero when the starting and ending points are the same.

Conservative forces Gravitational (without air resistance) Spring (without friction)

Nonconservative forces Depend on path Are not reversible Cannot be expressed in terms of a potential energy function Do work even when the starting and ending points are the same

Nonconservative forces Friction Chemical reaction forces

Nonconservative forces Increase or decrease the internal energy of a system Frequently, this happens through heat.

Law of conservation of energy This is always true – the most general form of the equation.

Power Power is not the same as energy or force. Power is the time rate at which work is done.

Power unit The unit of power is the watt

The kWh Your electric bill tells you how many kWh of electricity you used over the last month. This is a unit of energy or work, not a unit of power

Power in terms of velocity and force

Example Force A does 5 J of work in 10 s. Force B does 3 J of work in 5 s. Which force delivers greater power?

On your own A 5-kg box is being lifted upward at a constant velocity of 2 m/s by a force equal to the weight of the box. What is the power input of the force? How much work is done by the force in 4 s?