Chapter 5 Work and Energy

Concept Check - Work Is it possible to do work on an object that remains at rest? 1. yes 2. no

Concept Check - Work Is it possible to do work on an object that remains at rest? 1. yes 2. no Work requires that a force acts over a distance. If an object does not move at all, there is no displacement, and therefore no work done on that object. Does that mean that no work is being done?

Work Isometric exercises do work on the muscle tissue itself without doing work on an external object.

Concept Check – Play Ball!! In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by the catcher on the ball? 1. catcher has done positive work 2. catcher has done negative work catcher has done zero work

Concept Check – Play Ball!! In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by the catcher on the ball? 1. catcher has done positive work 2. catcher has done negative work catcher has done zero work The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F d cos q ), since  = 180o, then W < 0. Note that because the work done on the ball is negative, its speed decreases.

Concept Check – Work and Tension A ball tied to a string is being whirled around in a circle. What can you say about the work done by tension? 1. tension does no work at all 2. tension does negative work 3. tension does positive work v T

Concept Check – Work and Tension A ball tied to a string is being whirled around in a circle. What can you say about the work done by tension? 1. tension does no work at all 2. tension does negative work 3. tension does positive work No work is done because the force acts in a perpendicular direction to the displacement. Or using the definition of work (W = F d cos q ), since  = 90°, then W = 0. v T Follow-up: Is there a force in the direction of the velocity?

Concept Check – Work and Friction A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction? 1. friction does no work at all 2. friction does negative work 3. friction does positive work Fk Fn Fg displacement Fa

Concept Check – Work and Friction A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction? 1. friction does no work at all 2. friction does negative work 3. friction does positive work Friction acts in the opposite direction to the displacement, so the work is negative. Or using the definition of work (W = F d cos q ), since  = 180o, then W < 0. Fk Fn Fg displacement Fa

Work = Force × displacement Definition of Work In order for mechanical work to be done: Force must act on object Object must move* Work is a scalar, but it can be + or – (+) if Force and motion are in same direction (–) if Force and motion are in opposite directions Many forces can do work on the same object. Even friction can do work. Fn and any other force that is  to the direction of the motion will do no work. Only forces and components that are ∥ to the motion do work. The simplified form for the equation for work is: Work = Force × displacement * Sometimes it appears that no work is being done when it really is, like when you do isometric exercises.

Definition of Work Work is done on an object when a force causes a displacement of the object. Work is done only by components of a force that are parallel to a displacement. Work is equal to the product of force and the displacement it produces and the cosine of the angle between their directions.

Sample Fg = 1.50 x 103 N Fa = 345 N d = 24.0 m mk = 0.220 Wa = ? Wk = ? Wn= ? Wg = ? SW = ? Which ends up in the form of KE

Problem – 1 Fg = 5.00 x 109 N Ft = 5.00 x 103 N d = 3.00 km = 3.00 x 103 m Wt = ?

Problem – 3 Fa = 35 N θ = 25° d = 50.0 m Wa = ?

Problem – 4 Fg Fa d q Wa = ? Wk = ? Wn = ? Wg = ? mk = ?

Mechanical Energy Mechanical Energy Work is the transfer of Mechanical Energy from one form to another. Mechanical Energy Kinetic Energy (energy of motion) Potential Energy (energy of position) Gravitational Elastic 𝐾𝐸= 1 2 𝑚𝑣 2 𝑃𝐸 𝑔 =𝑚𝑔ℎ 𝑃𝐸 𝑒 = 1 2 𝑘𝑥 2

Kinetic Energy Kinetic energy is energy of motion. It depends on mass and speed. Kinetic Energy The energy of an object that is due to the object’s motion is called kinetic energy.

Kinetic Energy (Units) The energy of an object that is due to the object’s motion is called kinetic energy. These are the same units as for Work! Coincidence? I think not.

Gravitational Potential Energy PEg = mgh gravitational PE = mass  free-fall acceleration  height Potential Energy is the energy associated with an object because of the position, shape, or condition of the object. Gravitational potential energy is the potential energy stored in the gravitational fields of interacting bodies. Gravitational potential energy depends on height from a zero level. Given the zero location defined in Diagram A, how much PE would the ball have at points A,B,C,D, and E?

Elastic Potential Energy Elastic potential energy is the energy available for use when a deformed elastic object returns to its original configuration. The symbol k is called the spring constant, a parameter that measures the spring’s resistance to being compressed or stretched.

Springs in a car

Energy v. Time for Bouncing Ball When we say that something is conserved, we mean that it remains constant

Conservation of Mechanical Energy Mechanical energy is the sum of kinetic energy and all forms of potential energy associated with an object or group of objects. ME = KE + PEg + PEe Mechanical energy is conserved (remains constant) in the absence of friction even though the individual forms may change and acceleration may not be constant. MEi = MEf (KE + PEg + PEe)i = (KE + PEg + PEe)f The mathematical expression for the conservation of ME depends on the forms of potential energy given in a problem.

Sample h = hi = 3.00 m m = 25.0 kg vi = 0.0 m/s hf = 0 m vf = ? (Top) (Bottom)

Conservation of Mechanical Energy While individual forms of energy may change, the total mechanical energy remains constant (conserved) if there is… …little or no friction (including air resistance) …no collision in which significant heat is released Procedure: 1. Determine the forms of energy present (KE, PEg, PEe) . 2. Write expressions for the various energies at the beginning of the motion (KE + PEg + PEe)i and at the end (KE + PEg + PEe)f , where 𝐾𝐸= 1 2 𝑚 𝑣 2 , 𝑃 𝐸 𝑔 =𝑚𝑔ℎ, and𝑃 𝐸 𝑒 = 1 2 𝑘 𝑥 2 (identify when some terms will be zero). 3. Use the Law of Conservation of Energy (the total mechanical energy, ME, remains constant) to produce an equation that can be solved for the unknown.

Problem – 6 m hi g vf = ?

Problem – 8 hi m g vi vf = ?

Sample m k x vi = 0 vf = ? (Initial) (Final)

Finding h for a pendulum

Problem – 14 m k x g hf = ?

Work and Kinetic Energy A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? 1. positive work was done 2. negative work was done 3. zero work was done

Work and Kinetic Energy A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child? 1. positive work was done 2. negative work was done 3. zero work was done The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done. Or, from the Work-Kinetic Energy Theorem, W = DKE = KEf – KEi. Since we know that KEf > KEi in this case, then the work W must be positive. Follow-up: What does it mean for negative work to be done on the child?

Work and Kinetic Energy 𝐹 𝑎 If an object changes 𝐾𝐸, then Work was done on the object AND the net work done on the object EQUALS the change in 𝐾𝐸 Σ𝑊=∆𝐾𝐸 net work = change in kinetic energy This applies regardless of the nature of the forces doing work (i.e. friction) Work-Kinetic Energy Theorem The net work done by all the forces acting on an object is equal to the change in the object’s kinetic energy. The net work done on a body equals its change in kinetic energy. There are two ways to consider this relationship. The first is that one can find the net work done on an object by simply finding the change in kinetic energy of the object (KEf – KEi). Alternately, one can find the change in KE of an object by determining the net work done by adding together the work done by each force. Since work is a scalar, this is a scalar sum.

Sample Fa DKE d = ?

Sample m = 10.0 kg mk = 0.10 g = 9.81 m/s2 vi = 2.2 m/s vf = 0 m/s d = ? Which is the only force that will contribute to the net work?

power = work ÷ time interval Rate at which work is done, or rate at which energy is transferred. 𝑃= 𝑊 ∆𝑡 𝑃= 𝐹𝑑 ∆𝑡 =𝐹 𝑑 ∆𝑡 Power is a quantity that measures the rate at which work is done or energy is transformed. P = W/∆t power = work ÷ time interval An alternate equation for power in terms of force and speed is P = Fv power = force  speed 𝑃=𝐹𝑣

Simple Machines Actual Mechanical Advantage (AMA) is the ratio of how much force is applied by the machine to how much force is applied to the machine.

Simple Machines Ideal Mechanical Advantage (IMA) is the ratio of how far the effort moves the machine to how far the machine moves the load.

Simple Machines

Levers

Pulleys = 𝑑 𝑖𝑛 𝑑 𝑜𝑢𝑡

Inclines

Zipper

Wheel and Axle

Wheel and Axle

Sample The efficiency of a squeaky pulley system is _____ %. The pulleys are used to raise a mass to a certain height. What force is exerted on the machine if a rope is pulled _____ m in order to raise a _____ kg mass a height of _____ m? eff din dout m g Fin = ?