Section 8

Energy Methods Energy is the capacity of doing work. Energy methods are an analysis technique that results from the same fundamental equations as dynamic force analysis. Newton's Second law can be rewritten as: dv Σ F = ma = m dt dv ds dv ds dv = m   = m = mv dt  ds  ds dt ds Cross multiplying gives:

ΣF ds = m v dv Taking the anti-derivative:

ΣF s =

1 2 mv 2

which is a primary energy method equation. Energy methods are useful in analyzing machines that store energy. • counterweights • springs • flywheels

Work (U1? 2) Mechanical energy imposed onto, or removed from, an object during an interval between two states. •

Translating Objects

U1→ 2 = ∫ Fs ds ≈ Fs s Fs = force in the direction of motion s = displacement

•

Rotating Objects

U1→ 2 = ∫ Ts d θ ≈ Ts ∆ θ Ts = torque about the axis of rotation ∆θ = displacement

Once common form of energy loss is due to friction: U1? 2 = F f s = µ N s

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Potential Energy (PE) Amount of energy stored in an object. The object has the ability to convert its energy to another form, such as heat or motion. •

Due to Gravity PE = Wh W = weight of the object h = elevation of the object

•

Stored in a Spring PE = ½ kx2 k = spring constant or rate x = extension or compression of the spring

Kinetic Energy (KE) Amount of energy contained in a moving object. •

Translating Objects KE = ½ mv2 m = mass of the object v = linear velocity of the object

•

Rotating Objects KE = ½ Iω2 Ts = torque about the axis of rotation ω = angular velocity of the object

Conservation of Energy Energy is neither created nor destroyed; it is converted from form to form. This method of analysis consists of considering two states of the system and writing an energy equation that accounts for any energy transfer. ∆PE + ∆KE = U1→2 (PE2 - PE1 ) + (KE2 - KE1 ) = U1→2

Power (P) Analysis of machine components, such as motors and engines, requires not simply the quantity of work but also the time required to accomplish this work. Power is the rate of doing work

P=

dU ∆U ≈ dt ∆t

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Power (P) Unit conversions: ü 1 hp = 550 ft lb/s ü 1 Nm = 1 Joule (J) ü 1 Watt (W) = 1 Nm/s = 1 J/s ü 1 hp = 746 W •

Translating Objects

P= •

d ( F s) ds ≈F = Fv dt dt

Rotating Objects

P=

d(Tθ ) dθ ≈T = Tω dt dt

Efficiency (η) Energy conversion is not an ideal process. Some energy is lost to friction and generates heat.

η=

power output work output = power input work input

Hints for Solving Problems using Energy Methods: • • • •

•

When calculating work (U1→2 = Fs s), the force must be in the same direction as the distance. This may require multiplying the distance by a component of the force. The vertical height of an object is specified with respect to a fixed datum, which is usually the lowest point in the system. The distance that a spring is stretched or compressed is always with respect to its free length. The conservation of energy equation accounts for energy within a system and the energy transfer to and from the system. The following should be noted: ü The analysis is only concerned with the initial and final states. The intermediate conditions are of no consequence. ü To efficiently select the two states for the analysis, select one where everything about the system is known and another where a quantity must be determined. ü Energy can be lost from the system through friction or by driving a generator, turbine, etc. ü Energy can be introduced into the system by human force, or through a motor, engine, etc. Energy or power is always lost through a mechanical transmission due to friction. Use the efficiency of the system to account for these frictional losses.

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Problems: 8-1.

A cart is pushed 6 m along the slope by the 300 N force. Determine the work done on 300 the cart. 300 N 4Ans: 1559 N m

8-2.

100

A construction worker must apply the horizontal and vertical forces shown, while pushing the wheelbarrow 15 feet up the slope. Determine the work done.

60 lb 40 lb

4Ans: 872 ft lb

8-3.

200

Determine the work required to move the 100 lb crate, 10 ft along the conveyor. The coefficient of friction between the crate and conveyor is 0.4. 4Ans: 400 ft lb

8-4.

The crate has a mass of 20 kg and is pushed 3.5 m up the slope with the 130 N force. Determine the work done by the 130 N force and the work done by the friction force.

130 N

µk =0.15

4Ans: 455 & -89 N m 300

8-5.

A nut is tightened during its last half turn by an average torque of 20 Nm. Determine the work done. 4Ans: 63 N m

8-6.

A pipefitter maintains a 40 lb pull on a 12 in pipe wrench as she screws a fitting onto a pipe. How much work has been done when the wrench has turned 1800 . 4Ans: 1508 in lb

8-7.

In a rack and pinion assembly, a 50 ft lb torque is required to rotate the pinion over a quarter revolution. Determine the work (effort) to rotate the pinion. 4Ans: 79 ft lb

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. 4Ans: 239 ft lb

8-9.

Estimate the work required to punch a 0.75 diameter slug from a 0.10 in steel sheet. The steel has a shear strength of 25 ksi. 4Ans: 589 in lb

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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. 4Ans: 43200 ft lb 8-11. A jackscrew is used to raise a 6000 lb load. The screw uses an ACME 1.00 - 5 thread, which has 5 threads per inch. If the screw is rotated 25 revolutions, determine the increase of potential energy of the load. 4Ans: 2500 ft lb 8-12. A hole measuring10ft x 8 ft x 6 ft deep is to be excavated. Determine the amount of work done in bringing the soil to the original ground surface. The specific weight of the soil is 70 lb/ft3 . 4Ans: 100,800 ft lb 8-13. A circular water tower is filled by pumping water from a reservoir, whose surface remains at a constant level. Determine the total increase in potential energy of water, once the tank is filled. Water weighs 62.4 lb/ft3 .

8 ft

8 ft

18 ft

4Ans: 552,036 ft lb

8-14. Determine how much a spring, with a spring rate of 4 lb/in, will be compressed by a 13 lb force. Also, determine the stored energy in that compressed that spring. 4Ans: 21 in lb

8-15. A 125 lb force is required to compress a spring 4 in. Determine the stored energy in the compressed spring. 4Ans: 250 in lb 8-16. A spring, with a constant of 125 lb/in is initially extended 0.25 inches. The spring is then extended ano ther 0.75 inches. Determine the additional amount of stored energy in the spring. 4Ans: 59 in lb 8-17. A spring is initially compressed 50 mm by a force. If the force is increased by 250N, thereby causing the total spring compression of 150 mm, determine the spring constant. Also, determine the additional energy stored in the compressed spring. 4Ans: 25 N m 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 in. Also, determine the stored energy as it is compressed two additional inches from 3 to 5 inches. 4Ans: 1600 in lb 8-19. Determine the kinetic energy of a crate weighing 9 lb and traveling at 10 ft/s. 4Ans: 14 ft lb

8-20. Determine the kinetic energy of a truck weighing 9000 lb and traveling at 35 mph. How high would the equivalent amount of energy lift the truck? 4Ans: 368,261 ft lb

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8-21. By the time a starting pitcher is “pulled” in the eighth inning, he has thrown 350 pitches, including warm- up. If the ball weighs 0.32 lb and reaches 80 mph on each pitch, how much work has been done on the baseball? If he weighs 180 lb, how high would he have to climb a latter to do the equivalent amount of work? 4Ans: 23,943 ft lb

8-22. A flywheel is used on a compressor that runs at 1800 rpm. The flywheel has a 12 in diameter flywheel, weighs 2 lb and has a radius of gyration of 4 in. Determine the kinetic energy of that flywheel. 4Ans: 1471 in lb 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.

1m

4Ans: 7.3 N m 2m

8-24. Determine the kinetic energy of a 2 m slender rod, with a mass of 10 kg. It is rotated at 20 rpm as shown. 4Ans: 29 N m

2m

8-25. The driver of a 3000 lb. car, moving at 40 mph, applies the brakes. The pavement is wet, with µ s = 0.4 and µk = 0.35, and the car begins to skid. Determine the distance required for the car to stop. 4Ans: 153 ft 8-26. A 100 lb. load is pulled 15 ft. along the ground by a winch. The average coefficient of friction between the load and the ground is 0.8. The winch drum has a diameter of 12 inches. Determine the work and average torque required to turn the cylinder. 4Ans: 480 in lb 8-27. 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. 4Ans: 3254 N m

dumb waiter counterweight

8-28. The 300 lb crate shown has a velocity of 15 ft/sec down the slope. The coefficient of friction between the crate and the slope is 0.8. Determine how far the block will slide along the slope before stopping. 4Ans: 18 ft

300

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8-29. A 30,000 kg railroad car starts from rest and coasts down a 1 percent (0.570 ) incline for a distance of 20 m. At the bottom of the incline, the railroad car is stopped by a bumper having a spring constant of 2400 kN/m. Determine the speed of the car at the bottom of the incline, and how far the spring will be compressed in stopping the car. 4Ans: 2 m/s & 0.22m 3 ft

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. 4Ans: 4158 ft lb

6 ft

8-31. Determine the energy required to raise the 500 lb garage door, 15 ft. The spring has a rate of 4 lb/in and is unstretched when the door is fully open. 4Ans: 6150 ft lb

8-32. The Charpy testing machine consists of a 19 lb weight on the end of a 27 in arm. The material sample to be tested is clamped at the bottom and the weight is raised, released and allowed to break the material. Determine the energy absorbed by the test material if the initial and final positions of the arm are as shown. 4Ans: 276 in lb

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27” 3” 600 100

8-33. A hand brake is used to stop a 8 in. diameter disk, weighing 100 lb. The disk is initially rotating at 200 rpm, when a constant force of 10 lbs is applied to the handle. Determine the number of revolutions that the disk will turn after the force is applied. The friction between the disk and the brake is µs = 0.4 and µk = 0.3. 4Ans: 2.86 rev ω 8 in

10 lbs

2 in 8 in 18 in

8-34. A belt sander has a belt speed of 600 fpm. The coefficient of friction between new, 100 grit, sandpaper and wood is 0.5. Determine the power required to operate the sander when a user presses down with 30 lbs while sanding. 4Ans: 203 W

8-35. A propeller on a boat uses 22 hp to drive the boat at a speed of 16 mph. Determine the thrust force created by the propeller. 4Ans: 516 lb 8-36. A winch 2 ft diameter lifts a 3000- lb weight at 10 ft/sec. Determine the horsepower and torque that the winch must supply. 4Ans: 54.5 hp 8-37. A diesel engine has a maximum rated torque of 175 lb-ft when its power output is 50 hp. Determine the engine rpm at this operating condition. 4Ans: 1500 rpm 8-38. A flexible coupling is rated for 5 hp, when used on a 1200 rpm shaft. Determine the allowable torque for the coupling. 4Ans: 263 in lb 8-39. A punch press is to punch holes up to 1 in diameter in a steel plate, up to 0.125 in thick, with a shear strength of 50 ksi. The press is to be driven by an induction motor, with a rated speed of 1700 rpm. The allowable drop in motor speed is 15%. Find an appropriate flywheel inertia, and the motor power required for the press. 4Ans: 10.5 hp 8-40. In order to operate a fully- loaded, 2000 ft conveyor at 4 ft/sec requires 35 hp. Each linear foot of the conveyor caries a load of 20 lb. Determine the horsepower required if the conveyor is raised at one end, giving a gradient of 1 ft of rise in 60 ft of horizontal run. 4Ans: 40 hp 8-41. A winch 2 ft in diameter lifts a 200 lb weight at 20 ft/s. Determine the horsepower and torque that must be supplied to the drum. Also, determine the electrical power that must be supplied to a motor with an efficiency of 82%. 4Ans: 1800 in lb

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8-42. A jack hammer uses a rotary hydraulic motor to turn a cylindrical Splined shaft cam. The cam has a ramp, which causes the follower to compress a Cam follower 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 Compression spring 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. 4Ans: 4.6 hp

Hydraulic Motor

Stationary cam

Hammer

Impactor bit

8-43. Determine the power needed for a winch to haul a 100 kg crate up a 300 incline at a constant rate of 2 m/s. The friction between the crate and the inclined surface is is µs = 0.45 and µk = 0.4. 4Ans: 16.6 kW 8-44. In a forging device, a 40 kg hammer is lifted to an upper position shown. It is released, falls and strikes a workpiece. The spring rate is 1500 N/m, and the spring is stretched 100 mm, when the hammer is in the lower position. Calculate the impact velocity of the hammer. Also, determine the power needed to operate the machine if it delivers1 impact per second.

400 mm

300 mm workpiece

4Ans: 277 W

8-45. A 0.75 in hole is to be punched in a 0.125 in thick steel plate, with a shear strength of 30 ksi. The driving motor runs at 1200 rpm, and is geared to a countershaft which runs at 160 rpm, upon which a flywheel is mounted. The countershaft is, in turn, geared to the crankshaft of the press, which gives 20 punching operations per minute. Determine the power required to drive the press, if no flywheel is used. Then, determine an appropriate flywheel inertia, if no more than 10% speed fluctuation is permitted. Finally, determine the power required to operate the press.

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8-46. A pile-driver hammer weighing 1100 lb is raised vertically a distance of 10 ft in 5 sec, at approximately constant velocity. Determine the power required from the engine, if the hoisting equipment is 80% efficient. 4Ans: 5 hp 8-47. A utility hoist can lift a 3000 kg crate at a rate of 25 m/min. Knowing that the hoist is run by a 15 kW motor, determine the overall efficiency. 4Ans: 81.7% 8-48. The motor of a hoist shown operates with an efficiency of 85%. Determine the power that must be supplied to the motor to lift the 75 lb. crate at a rate of 2 ft/s. 4Ans: .32 hp

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

4Ans: 6.3 kW

8-50. The 50 kg crate is hoisted up the 300 incline by a motor as shown. The crate must travel 8 m along the plane in 2 sec. Determine the power that must be supplied to the motor. The coefficient of friction between the crate and incline is 0.3. The motor efficiency is 84% 8-51. A continuous-belt bucket elevator raises grain a vertical distance of 70 ft before discharging it at the top of a bin. The efficiency of the elevator is 75% and the driving motor is 90% efficient. If electricity costs 8 cents per kilowatt- hour, determine the cost per ton of grain. 8-52. The escalator is designed to transport 6000 persons per hour at a constant speed of 1.5 ft/s. Assuming that the average weight of a 150 lb person, determine the average power required if the mechanical efficiency is 85 % and if a 300 percent overload is to be allowed.

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15 ft

300