Energy is the capacity of doing work.

Energy methods are useful in analyzing machines that store energy.

counterweights springs flywheels

Energy MethodsSection 8

Energy Methods from Newton

ds

dvmv

dt

ds

ds

dvm

ds

ds

dt

dvm

dt

dvmmaF

dvvmdsF

2

2

1mvsF

where:s = displacementv = velocitya = acceleration

Mechanical energy imposed onto an object between two states (locations).

Translating Objects:

Fs = force in the direction of motion

s = displacement

sFdsFU ss21

Linear Work (U12)

Problem 8-1:

A cart is pushed 6 m along the slope by the 300 N force. Determine the work done on the cart.

300

300 N

100

Mechanical energy imposed onto an object between two states (angles).

Rotating Objects

Ts = torque about the axis of rotation

= displacement

Rotational Work (U12)

ss TdTU 21

Problem 8-8:

The flywheel on a punch press delivers 3000 ft-lbs of energy to punch a hole. The flywheel turns two revolutions, for every punch stroke. Determine the average torque that needs to be supplied to the punch press.

Potential Energy (PE)

Amount of energy stored in a stationary object.

Potential Energy due to gravity:

PE = Wh

W = weight of the object

h = elevation of the object

Problem 8-10:

An 1200 lb elevator moves from the 2nd floor to the 5th floor, a distance of 36 ft. Determine the increase of potential energy of the elevator.

Potential Energy (PE)

Energy Stored in a Spring

PE = ½ k x2

k = spring constant (rate)

x = deformation of the spring

Problem 8-18:

A coil spring is used on a bumper, and has a spring constant of 200 lb/in. Determine the amount of stored energy, as it is compressed the first 3 inches. Also, determine the additional stored energy as it is compressed two additional inches from 3 to 5 inches.

Kenetic Energy (KE)

Energy stored in a moving object.

Translating Objects:

KE = ½ mv2

m = mass of the object

v = linear velocity of the object

Problem 8-20:

Determine the kinetic energy of a truck weighing 9000 lbs and traveling at 35 mph. How high would the equivalent amount of energy lift the truck?

Kenetic Energy (KE)

Rotating Objects

KE = ½ I2

Ts = torque about the axis of rotation

= angular displacement

Problem 8-23:

Determine the kinetic energy of a 2 m slender rod, with a mass of 10 kg. It is rotated at 20 rpm as shown.

1 m

2 m

Conservation of Energy

Energy is neither created nor destroyed. It is converted from form to form.

PE + KE = U12

First Law of ThermodynamicsGeneral Energy Equation

Problem 8-26

A dumb waiter and its load have a combined weight of 600 lb. It is configured with an 800 lb counterweight as shown. Determine the energy (work) required from the motor to raise the dumb waiter 12 ft.

counterweight

dumb waiter

Problem 8-30

A lift that is used to dump the contents of a 55 gallon drum is shown below. The maximum density of the contents is12 lbs per gallon. Determine the work required to lift the container 6 ft. The coefficient of friction between the collars and the guide rod is 0.1

6 ft

3 ft

Power (P)

Rate of doing work

Units:1 hp = 550 ft lb/s1 Nm = 1 Joule (J)1 Watt (W) = 1 Nm/s = 1 J/s1 hp = 746 W

t

U

dt

dUP

Power

Translating Objects:

Rotating Objects:

Fvdt

dsF

dt

sFdP

)(

T

dt

dT

dt

TdP

)(

Problem 8-34

A belt sander has a belt speed of 400 fpm. The coefficient of friction of new, 100 grit paper is 0.5. Determine the power required to operate the belt when a user presses down with 30 lb while sanding.

Problem 8-38

A flexible coupling is rated for 5 hp, when used on a 1200 rpm shaft. Determine the allowable torque for the coupling

Problem 8-42

A jack hammer uses a rotary hydraulic motor with a cylindrical cam. The cam has a ramp, which causes the follower to compress a spring, and a step to release the follower. The spring has a rate of 2000 lb/in and the cam lift is 2 in. At the end of the stroke, the spring is compressed 0.5 inches. The hammer weighs 30 lbs. Determine the impact velocity of the hammer. Also, estimate the hydraulic power needed to drive the jackhammer if it delivers 5 strokes per second.

Problem 8-42 (con’t)

Stationary camCam follower

Hammer

Splined shaft

HydraulicMotor

Compression spring

Impactor bit

Efficiency ()

Energy conversion is not an ideal process. Some energy is lost with friction and generates heat.

inputwork

outputwork

inputpower

outputpower

Problem 8-49

Crushed stone is being moved from a quarry to a loading dock at a rate of 500 tons/hr. An electric generator is attached to the system in order to maintain constant belt speed. Knowing that the efficiency of the belt/generator system is 70%, determine the average killowatts developed by the generator. The belt speed is 10 ft/sec.

240 ft

3 mi