Physics 160 Biomechanics

Work, Energy and Power

Questions to Think About • In which direction should you apply a force in order to

generate the most power? • Why would a faster speed down the runway allow a

gymnast to vault higher? • Why would a pole vaulter need to switch to a stiffer

pole in order to vault higher?

Mechanical Work Mechanical work done by a force is the component of the force

in the direction of motion times the displacement of the object.

[ ] [ ][ ]

[ ]1 1

W F dW work in Joules JF force in Nd displacement in mJ N m

= ⋅= ==== ⋅

Work is a scalar quantity that can be positive, negative or zero.

Positive and Negative Work • Positive work: Force and displacement in same direction.

Indicates force acting to increase the object’s speed.

• Negative work: Force and displacement in opposite directions. Indicates force acting to decrease object’s speed.

• Zero work: Force and displacement are perpendicular. Indicates force acting neither speeds up nor slows down object.

Friction

Pushing force does positive work, friction does negative work, R and FW do zero work

Fw

R

Example A therapist is helping a patient with stretching

exercises. She pushes on the patient’s foot with an average force of 200 N. The patient resists the force and moves the foot 20 cm toward the therapist. How much work did the therapist do on the patient’s foot?

Example Bob bench presses a 50 lb

barbell. He maintains a constant speed on the way up and on the way down. If the barbell moves 75 cm in each direction how much work does Bob do on the barbell on the way up and the way down? What is the total work done?

Power Power is the rate of doing

work or how much work is done in a given amount of time.

Power is a scalar quantity

and can be positive, negative or zero.

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[ ][ ]

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W F dP F vt t

P power in Watts WW work in Jt time in sF average force in Nv average velocity in m sW J shp W

⋅= = = ⋅

= ==

=====

Example A 75 kg person runs up a flight

of 30 stairs of riser height of 25 cm during a 20 s period. How much mechanical work is done? How much mechanical power is generated?

Muscular Power

Muscular power is the product of muscular force and the velocity of muscle shortening.

Forc

e

Velocity

Pow

er

Power-Velocity Force-Velocity

Maximum power occurs at approximately one-third of maximum velocity and at approximately one-third of maximum concentric force.

Muscular Power The relationship between force and velocity in eccentric muscle

action is opposite to that of concentric muscle action. In eccentric muscle action, the force increases as the velocity of the lengthening increases. The force continues to increase until the eccentric action can no longer control lengthening of the muscle.

Example An Olympic weightlifter snatches 100 kg. In a snatch,

the barbell is moved from a stationary position on the floor to a stationary position over the athlete’s head. Only 0.50 s elapsed from the first movement of the barbell until it was overhead, and the barbell moved through a vertical displacement of 2.0 m. What was the weightlifter’s average power output during the lift?

Kinetic Energy Energy is defined as the capacity to do work. Kinetic energy is the energy of motion.

2

2 2

1. .2

. . [ ][ ][ / ]

1 1 /

K E m v

K E kinetic energy in Jm mass in kgv speed in m sJ kg m s

= ⋅

===

= ⋅

Kinetic energy is a scalar that can only be positive or zero.

Example Compare the kinetic energies of the following: a) A baseball (mass=0.145 kg) moving at 100 km/h. b) A runner (mass=75 kg) moving at 30 km/h. c) A swimmer (mass=75 kg) moving at 7 km/h. d) A discus (mass=2 kg) moving at 20 m/s.

Potential Energy Potential energy is stored energy due to position.

2

. .

. . [ ]

[ ]9.8 /

[ ]

g

g

P E mghP E gravitational potential energy in Jm mass in kgg m sh height above reference height in m

=

=

=

==

Gravitational P.E. is a scalar and can be positive, negative or zero depending on the choice of the zero reference height.

Example Compare the gravitational potential energies of the

following: (let the ground be h=0) a) A 70 kg pole vaulter at the top of a 5.9 m bar. b) A 50 kg gymnast in a giant swing 3.5 m above

ground.

Strain Energy Strain energy (or elastic energy) is potential energy

due to the deformation of an object.

21. .2

. . [ ][ / ]

[ ]

s

s

P E kx

P E strain energy in Jk spring constant in N mx change in length or deformation in m

=

===

Strain energy is a scalar and is positive or zero.

Example Compare the strain energy stored in the following: a) A tendon that is stretched 5 mm if the stiffness of

the tendon is 10,000 N/m. b) A diving board that is bent down 0.8 m if the

effective spring constant is 833 N/m.

Conservation of Energy

The total energy of a closed system is constant. Energy can neither be created nor destroyed, only

converted from one form into another. ( . . . . . . ) ( . . . . . . )

. .. .

. .

g s initial g s final

g

s

K E P E P E K E P E P EK E kinetic energyP E gravitational potential energyP E strain potential energy

+ + = + +

==

=

Conservation of Energy

Changes in potential energy (PE) and kinetic energy (KE) as a ball is projected straight up and as it falls back to earth.

Example

A volleyball is bumped vertically up to a height of 10 m above the player’s arms. What was the initial velocity of the ball just after leaving the player’s arms?

Example A 50 kg pole vaulter has a horizontal velocity of 8 m/s

at the completion of her approach run, and her center of gravity is 1.0 m high. Estimate how high she should be able to vault if her kinetic and potential energies were all converted to potential energy.