C-10 Work and Energy Physics Jan. 2015

10.1 Machines and Mechanical Advantage The ability of humans to build buildings and move mountains began with our invention of machines. In physics the term “simple machine” means a machine that uses only the forces directly applied and accomplishes its task with a single motion. 2

10.1 Machines The best way to analyze what a machine does is to think about the machine in terms of input and output. 3

10.1 Mechanical Advantage Mechanical advantage is the ratio of output force to input force. For a typical automotive jack the mechanical advantage is 30 or more. A force of 100 newtons (22.5 pounds) applied to the input arm of the jack produces an output force of 3,000 newtons (675 pounds)— enough to lift one corner of an automobile.

10.1 Mechanical Advantage MA = Fo Fi Output force (N) Mechanical Input force (N)

10.1 Types of simple machines The lever, wheel and axle, rope and pulleys, screw, ramp, and gears are the most common simple machines. Complex machines, combine many simple machines into mechanical systems. A mechanical system is an assembly of simple machines that work together to accomplish a task.

10.1 Mechanical Advantage of a Lever The essential features of a lever are the input arm, output arm, and fulcrum.

10.1 Three types of levers The three types of levers are classified by the location of the input and output forces relative to the fulcrum: first class lever second class lever third class lever

10.1 How a lever works A lever works by rotating about its fulcrum. The mechanical advantage can be deduced by calculating the torques created by the input and output forces. The input force creates a (positive) counterclockwise torque. The torque created by the reaction force is clockwise (negative). When the lever is in equilibrium the net torque must be zero.

10.1 Torque and mechanical advantage

Calculating the position of the fulcrum Where should the fulcrum of a lever be placed so one person weighing 700N can lift the edge of a stone block with a mass of 500 kg? The lever is a steel bar three meters long. Assume a person can produce an input force equal to their own weight. Assume that the output force of the lever must equal half the weight of the block to lift one edge. You are asked for the location of the fulcrum You are given input force, lever length, mass to be lifted Since you are told to assume Fo = ½ Fw , use: Fw = mg , MA = Fo ÷ Fi , MA = Li÷ Lo Solve for Fo = (.5) (500 kg) (9.8 N/kg) = 2,450 N Solve for MA = 2,450 N ÷ 700 N = 3.5 Since Li = 3.5 Lo , and Li + Lo = 3m, then Lo = .67m

10.1 Mechanical advantage of ropes and pulleys A tension force is a pulling force acting along the direction of a rope or string. Ropes and strings carry tension forces throughout their length. If friction is small, the tension force in a rope is the same everywhere. If you were to cut a rope in tension and insert a force scale, the scale would measure the same force at any point along the rope.

10.1 Rope & Pulleys The block-and-tackle machine is a simple machine using one rope and multiple pulleys. The rope and pulleys can be arranged to create different amounts of mechanical advantage.

10.1 Wheels, gears, & rotating machines Wheels and axles provide advantages. Friction occurs where the wheel and axle touch or where the wheel touches a surface. Rolling motion creates less wearing away of material compared with two surfaces sliding over each other. With gears the trade-off is made between torque and rotation speed. An output gear will turn with more torque when it rotates slower than the input gear.

10.1 Ramps and Screws Ramps reduce input force by increasing the distance over which the input force needs to act. A screw is a simple machine that turns rotating motion into linear motion. A thread wraps around a screw at an angle, like the angle of a ramp. 16

10.2 Work A force acts upon an object to cause a displacement of the object. Forces make things move. (energy transfer) Work is only done when a force moves an object through a distance.

Work = Force x Distance We do work when we lift a load or push a box across the floor. The force is in the same direction as the displacement. Two things enter into every case where work is done 1. The application of a force 2. The movement of something in the same direction as that force.

Which of the following are examples of work? A person pushing a shopping cart through the store. YES. The force of the person causes the cart to get displaced. A weightlifter lifting a barbell above his head. YES. The force of the weightlifter displaces the barbell upward. A person applies a force to a wall and becomes exhausted. NO. The wall is not displaced. A book falls off a table and free falls to the ground. YES. The force of gravity acts on the book which causes it to be displaced in a downward direction (fall). You finish your physics homework before you leave class. NO. The is a not the scientific form of work, this is work done in an everyday sense!

Equations for WORK W = F x d Straight line force W = Fd cos () Distance (m) Force (N) Work (joules) W = F x d Straight line force W = Fd cos () Force acts at an angle Angle 

10.2 Work done against gravity Mass (kg) Height object raised (m) Work (joules) W = mgh Gravity (m/sec2) Remember that F = mg

Work Practice Problems: A 10.5 g hockey puck is sliding across the ice. A player exerts a constant force of 4.5 N over a distance of 0.15 m. How much work does the player do on the puck? W=Fxd W = 4.5 N x 0.15 m = __________? 0.675 J

Work Practice Problem A sailor pulls a boat 30 m along a dock using a rope that makes a 25° angle with the horizontal. How much work does the sailor do on the boat if he exerts a force of 255N on the rope? W = Fd cos(θ) W = 255N x 30m (cos 25°) = _______? 6933J

10.2 Work done by a machine Work is usually done when a force is applied to a simple machine. All machines can be described in terms of input work and output work. In any machine, some of the input work goes to overcoming friction. The output work is always less than the input work because of the energy lost to friction.

ENERGY Sections 10.3 and 11.3

ENERGY Energy describes a system’s ability to cause change. A system that has energy has the ability to do work. Energy is measured in the same units as work because energy is transferred during the action of work.

MECHANICAL ENERGY Kinetic Energy (motion) Mechanical energy is the total energy possessed by an object due to its motion or its position. Kinetic Energy (motion) Potential Energy (position) Gravitational Elastic ME = KE + PEg + PEe

Potential Energy (PE) Definition: energy associated with an object due to its position. An object has the potential to move and change its position. It is a SCALAR quantity, it does not have a direction. SI unit for PE is also the Joule (N•m). Two types of PE: Gravitational PEg Elastic PEe

Gravitational PE (PEg) Definition: The energy associated with an object due to the object’s position relative to a gravitational source. PEg depends on an object’s MASS, HEIGHT, and free-fall acceleration (GRAVITY). Equation: m is mass of the object (kg) g is the free-fall acceleration (m/s2) h is the height (m) PEg = mgh

Practice Problem #1 - Potential Energy A cart with a mass of 200 kg is pushed up a ramp. The top of the ramp is 4 meters higher than the bottom. How much potential energy is gained by the cart?

PEg = mgh m= 200kg g = 9.8 m/s2 h = 4 m PE = (200 kg)(9.8 m/s2)(4 m) = 7840 J

Elastic Potential Energy (PEelastic) Definition: the PE stored in any compressed or stretched elastic object (springs, strings) When external forces compress or stretch the spring, elastic potential energy is stored.

The amount of energy depends on the distance the spring is compressed or stretched from its relaxed length and its spring constant.

PEelastic PE(elastic) = ½ k x2 Equation: k is a symbol for the spring constant (N/m) x is the distance compressed or stretched (m). Remember: spring constant is the force constant or how resistant a spring is to being stretched or compressed. PE(elastic) = ½ k x2

Practice Problem #2 - Potential Energy A 70 kg stuntman is attached to a bungee cord with an unstretched length of 15m. He jumps off a bridge and when he finally stops, the cord has a stretched length of 44 m. The spring constant of the bungee cord was calculated to be 71.8N/m. What is the elastic potential energy of the bungee cord?

Kinetic Energy (KE) KE = ½ mv2 Definition: The energy of an object due to its motion KE depends on both an object’s MASS and its SPEED. It is a SCALAR quantity, it does not have a direction. SI unit for KE is the Joule (N • m). Equation: m is the mass (kg) v represents speed, not velocity (m/s) KE = ½ mv2

Pracitce Problem #3&4 – Kinetic Energy A 7.0 kg bowling ball moves at 3.0 m/s. How much kinetic energy does the bowling ball have? KE = ½ (7.0kg)(3.0m/s)2 = 31.5 J How fast must a 2.45 g table-tennis ball move in order to have the same kinetic energy as the bowling ball?

Work – Kinetic Energy Theorem When work is done by external forces (push, pull, friction, etc), the energy of the object is altered. The net work done on an object is equal to the change in kinetic energy of the object. If the force and the displacement are in the same direction, then positive work is done on the object and the object gains KE. If the force and the displacement are in the opposite direction, then negative work is done on the object; the object loses KE.

Is Work + or - ? Change in KE? Megan drops the ball and hits it with her tennis racket. The racket is moving horizontally as the strings apply a horizontal force while in contact with the ball. Work is Positive KE increases (ball gains speed) The frictional force between highway and tires pushes backwards on the tires of a skidding car. Work is Negative KE decreases (car slows down)

Work – Kinetic Energy Theorem Equation:

Practice Problem #5 A 1150 kg truck speeds up from 15 m/s to 30 m/s while passing another truck. How much work was done on the truck to increase its speed? ∆KE = KEf - KEi KEf = 1/2(1150kg)(30m/s)2 =517500J KEi = 1/2(1150kg)(15m/s)2 =129375J ∆KE = 517500J - 129375J = 388000 J

Law of Conservation of Energy No new energy is created and no existing energy is destroyed. As energy takes different forms and changes things by doing work, nature keeps perfect track of the total.

Conservation of Mechanical Energy Mechanical energy (ME) is the total of the kinetic and potential energies acting on an object. ME is conserved when the only type of force doing work upon an object is an internal force (gravity or spring forces). The object’s energy changed from: PE →KE or KE → PE.

How is Mechanical Energy conserved How is Mechanical Energy conserved? Identify the change in Energy & Internal force. A ball falls from a height of 2 meters in the absence of air resistance. Change in PE to KE Internal force: gravity A bungee cord begins to exert an upward force upon a falling bungee jumper. Change in KE to PE Internal force: spring force http://www.physicsclassroom.com

The Example of Pendulum Motion A pendulum bob is swinging to and fro on the end of a string. There are only two forces acting upon the pendulum bob. Gravity (an internal force) acts downward Force of Tension(an external force) pulls upwards towards the pivot point. The external force does not do work since it is perpendicular to the motion.

As the height decreases, PE is lost; and KE is gained. As the pendulum bob swings, its height above the table top and in its speed is constantly changing. As the height decreases, PE is lost; and KE is gained. At all times, the SUM of the PE and KE of the bob remains constant. There is NO loss or gain of mechanical energy; only a transformation from KE to PE (and vice versa). 100 TIME (s) E N E R G Y (J) PEg KE http://www.physicsclassroom.com/mmedia/energy/pe.cfm

Think about this carefully – you will see it again.

The Example of a Roller Coaster http://www.physicsclassroom.com/mmedia/energy/ce.cfm

Conservation of ME Equations Mechanical energy is the total of the kinetic and potential energies acting on an object. The initial mechanical energy is equal to the final mechanical energy, in the absence of friction.