# Chapter 5 – WORK and ENERGY. 5.2 MECHANICAL ENERGY.

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Chapter 5 – WORK and ENERGY

5.2 MECHANICAL ENERGY

ENERGY = ability to do WORK

the energy of an object that is due to the object’s motion KINETICPOTENTIAL the energy associated with an object because of its position, shape, or condition

KINETIC ENERGY A cart: mass = m force = constant So a = constant ΔxΔx W net = F ∙ d W net = F ∙ Δx W net = m ∙ a ∙ Δx

KINETIC ENERGY ΔxΔx W net = m ∙ a ∙ Δx W net = ½ mv f 2 – ½ mv i 2

KINETIC ENERGY depends on SPEED and MASS KINETIC ENERGY = KE = E k = W net = ½ mv f 2 – ½ mv i 2

KINETIC ENERGY depends on SPEED and MASS KINETIC ENERGY = KE = E k = W net = ½ mv f 2 – ½ mv i 2 = ΔKE = ΔE k

WORK – KINETIC ENERGY THEOREM Net work = change in kinetic energy W net = ½ mv f 2 – ½ mv i 2 W net = KE f – KE i W net = ΔKE

KINETIC ENERGY = KE = E k = UNITS = Joules KINETIC ENERGY depends on SPEED and MASS

WORKSHEET EXAMPLE A 6.0 kg cat runs after a mouse at 10 m/s. What is the cat’s kinetic energy?

WORKSHEET EXAMPLE A 6.0 kg cat runs after a mouse at 10m/s. What is the cat’s kinetic energy? KE = 300 J

W net = 4.1975 x 10 4 J = 4.2 x 10 4 J W net = ½ mv f 2 – ½ mv i 2 = ΔKE W net = 7.2 x 10 - 2 J

EXAMPLE On a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10 ?

EXAMPLE On a frozen pond, a person kicks a 10.0 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10 ? GIVEN? UNKNOWN? FBD? mass = 10.0 kg v i = 2.2 m/s v f = 0.0 m/s μ k = 0.10 d = Δx = ??

EXAMPLE mass = 10.0 kg v i = 2.2 m/s v f = 0.0 m/s μ k = 0.10 d = Δx = ?? W net = F ∙ d Net W net is due to F f, so… W net = F f ∙ d -½ mv i 2 = μ k ∙ m ∙ g ∙ d d = 2.5 m

WORKSHEET EXAMPLE How much net work is needed to accelerate a 1000.0 kg car from 20.0 m/s to 30.0 m/s

WORKSHEET EXAMPLE How much net work is needed to accelerate a 1000.0 kg car from 20.0 m/s to 30.0 m/s

HOMEWORK Worksheet Problems: Practice B Practice C 19

POTENTIAL ENERGY gravitational elastic

POTENTIAL ENERGY the energy associated with an object because of its position, shape, or condition It is stored energy

POTENTIAL ENERGY Depends on the properties of an object and its interactions with its environment It is stored energy

GRAVITATIONAL POTENTIAL ENERGY Is the energy of an object due to the objects position relative to a gravitational source i.e. energy stored in an object due to its position relative to the Earth’s gravitational field

GRAVITATIONAL POTENTIAL ENERGY i.e. energy stored in an object due to its position relative to the Earth’s gravitational field Potential energy Kinetic energy

GRAVITATIONAL POTENTIAL ENERGY i.e. energy stored in an object due to its position relative to the Earth’s gravitational field Potential energy Kinetic energy

GRAVITATIONAL POTENTIAL ENERGY i.e. energy stored in an object due to its position relative to the Earth’s gravitational field UNITS = Joules REMEMBER: a.free – fall acceleration = constant (near the E’s surface) for PE g to be valid b.It is relative (because measuring the height is arbitrary)

TRUE or FALSE? Suppose I throw a volleyball from a second floor roof (A), and it lands on the first floor of an adjacent building (B)… If the height is measured from the ground (C), PE g is zero. If the height is measured from the 1 st floor (B), PE g is zero.

Can You Have a Negative PE? If YES, give an example If NOT, explain why.

Can You Have a Negative PE? You could bring a textbook from a table height to a zero – level (ground) – performing a negative work If yes, give an example If not, explain why.

Can you have an object with a positive PE relative to one point and negative PE to another point at the same time? If YES, give an example If NOT, explain why.

A textbook which is 0.5 m below a table (negative PE relative to the table) If yes, give an example If not, explain why. A textbook which is 0.5 m above the ground (positive PE relative to the table) Can you have an object with a positive PE relative to one point and negative PE to another point at the same time?

m = 25 kg h = 6 m Potential energy doubles

ELASTIC POTENTIAL ENERGY Is the energy of available for use when a deformed elastic object (spring, bungee cord…) returns to its original position x

k = SPRING CONSTANT Measures how easy is to compress or stretch a spring Very flexible Easy to stretch Very stiff Difficult to stretch

WHAT IS THE UNIT OF k?

WORKSHEET EXAMPLE as a point mass A 70.0 kg stuntman is attached to a bungee cord with an unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord and air resistance. Assuming the spring constant of the bungee cord is 71.8 N/m, WHAT IS THE TOTAL POTENTIAL ENERGY RELATIVE TO THE WATER WHEN THE MAN STOPS FALLING?

EXAMPLE as a point mass A 70.0 kg stuntman is attached to a bungee cord with an unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord and air resistance. Assuming the spring constant of the bungee cord is 71.8 N/m, WHAT IS THE TOTAL POTENTIAL ENERGY RELATIVE TO THE WATER WHEN THE MAN STOPS FALLING? PE tot = PE g + PE elastic

EXAMPLE as a point mass A 70.0 kg stuntman is attached to a bungee cord with an unstretched length of 15.0 m. He jumps off a bridge spanning a river from a height of 50.0 m. When he finally stops, the cord has a stretched length of 44.0 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the bungee cord is 71.8 N/m, What is the total potential energy relative to the water when the man stops falling? GIVEN?UNKNOWN?FBD? Choose the water level to be 0 PE g

EXAMPLE mass = 70.0 kg k = 71.8 N/m h = 50.0 – 44.0 = 6.0 m x = 44.0 – 15.0 = 29.0 m PE = 0 J (at river level) PE tot = PE g + PE elastic = ? h Choose the water level to be 0 PE g x

EXAMPLE mass = 70.0 kg k = 71.8 N/m h = 50.0 – 44.0 = 6.0 m x = 44.0 – 15.0 = 29.0 m PE = 0 J (at river level) PE tot = PE g + PE elastic = ? PE g = m ∙ g ∙ h PE elastic = ½ kx 2 PE tot = 3.43 x 10 4 J

WORKSHEET EXAMPLE When a 2.00 kg mass is attached to a vertical spring, the spring is stretched 10.0 cm such that the mass is 50.0 cm above the table. a.What is the gravitational potential energy associated with this mass relative to the table? b.What is the spring’s elastic potential energy if the spring constant is 400.0 N/m ? c.What is the total potential energy of this system?

WORKSHEET EXAMPLE When a 2.00 kg mass is attached to a vertical spring, the spring is stretched 10.0 cm such that the mass is 50.0 cm above the table. a.What is the gravitational potential energy associated with this mass relative to the table? b.What is the spring’s elastic potential energy if the spring constant is 400.0 N/m ? c.What is the total potential energy of this system?

HOMEWORK Worksheet: Problems: Practice D 43

MECHANICAL ENERGY Chemical, nuclear, electrical… ME = KE + PE

MECHANICAL ENERGY PENDULUM

MECHANICAL ENERGY Describe each picture in terms of potential and kinetic energy

MECHANICAL ENERGY PE g present (at max) No KE present PE g present KE present No PE g present KE present (at max) PE g present (at max) No KE present PE g present KE present No PE g present KE present (at max)

MECHANICAL ENERGY

ME = KE + ∑ PE

CONSERVATION OF ENERGY Who can describe what is happening in the next demonstration?

ME is often conserved  E remains constant KE i + ∑ PE i = KE f + ∑ PE f

CONSERVATION OF ENERGY KE i + ∑ PE i = KE f + ∑ PE f

CONSERVATION OF ENERGY

EXAMPLE Starting from rest, a child zooms down a frictionless slide from an initial height of 3.00 m. WHAT IS HER SPEED AT THE BOTTOM OF THE SLIDE? Assume she has a mass of 25.0 kg.

EXAMPLE Starting from rest, a child zooms down a frictionless slide from an initial height of 3.00 m. WHAT IS HER SPEED AT THE BOTTOM OF THE SLIDE? Assume she has a mass of 25.0 kg. GIVEN?UNKNOWN? FBD? v i = 0.0 m/s h = h i = 3.00 m h f = 0.00 m v f = ? m/s

EXAMPLE Starting from rest, a child zooms down a frictionless slide from an initial height of 3.00 m. WHAT IS HER SPEED AT THE BOTTOM OF THE SLIDE? Assume she has a mass of 25.0 kg. FBD? PE g,i = m ∙ g ∙ h i PE g,f = 0 J KE i = 0 J KE f = ½ mv f 2 KE i + ∑PE i = KE f + ∑PE f v f = 7.67 m/s

WORKSHEET EXAMPLE A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0 × 10 2 N/m.

WORKSHEET EXAMPLE A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0 × 10 2 N/m. a. What is the elastic potential energy of the slingshot before it is released? b. What is the kinetic energy of the ball just after the slingshot is released? c. What is the ball’s speed at that instant? d. How high does the ball rise if it is shot directly upward?

WORKSHEET EXAMPLE A small 10.0 g ball is held to a slingshot that is stretched 6.0 cm. The spring constant is 2.0 × 10 2 N/m. a. What is the elastic potential energy of the slingshot before it is released? b. What is the kinetic energy of the ball just after the slingshot is released? c. What is the ball’s speed at that instant? d. How high does the ball rise if it is shot directly upward?

HOMEWORK Problems: Practice E 60

QUIZ Date: Tuesday, Dec 17 What?: Chapter 5.2 Mechanical Energy 61

WORKSHEET EXAMPLE A 6.0 kg cat runs after a mouse at 10m/s. What is the cat’s kinetic energy? KE = 300 J

WORKSHEET EXAMPLE How much net work is needed to accelerate a 1000.0 kg car from 20.0 m/s to 30.0 m/s

Can You Have a Negative PE? You could bring a textbook from a table height to a zero – level (ground) – performing a negative work If yes, give an example If not, explain why.

Can you have an object with a positive PE relative to one point and negative PE to another point at the same time? A textbook which is 1.0 m below a table (negative PE relative to the table) If yes, give an example If not, explain why. A textbook which is 1.0 m above the ground (positive PE relative to the table)

WHAT IS THE UNIT OF k? N/m

WORKSHEET EXAMPLE When a 2.00 kg mass is attached to a vertical spring, the spring is stretched 10.0 cm such that the mass is 50.0 cm above the table. a.What is the gravitational potential energy associated with this mass relative to the table? b.What is the spring’s elastic potential energy if the spring constant is 400.0 N/m ? c.What is the total potential energy of this system?