Download Physics 160 Biomechanics

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.
W= F ⋅ d
W work
=
in [ Joules ] [ J ]
F = force in [ N ]
d = displacement in [m]
1=
J 1N ⋅ 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.
Pushing force does
positive work, friction
does negative work, R
and FW do zero work
R
Friction
Fw
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
W F ⋅d
=
= F ⋅v
t
t
power
=
in [Watts ] [W ]
W = work in [ J ]
t = time in [ s ]
F = average force in [ N ]
v = average velocity in [m / s ]
1W = 1J / s
1hp = 746W
Power is the rate of doing
P=
work or how much work is
done in a given=
amount of
P
time.
Power is a scalar quantity
and can be positive,
negative or zero.
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.
Power
Maximum power occurs at
approximately one-third of
maximum velocity and at
approximately one-third of
maximum concentric force.
Force
Force-Velocity
Power-Velocity
Velocity
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.
1
.E .
K=
m ⋅ v2
2
K .E. = kinetic energy in [ J ]
m = mass in [ kg ]
v = speed in [ m / s ]
1
J
=
1kg ⋅ m 2 / s 2
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.
P.E.g = mgh
P.E.g = gravitational potential energy in [ J ]
m = mass in [kg ]
g = 9.8m / s 2
h = 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.
1 2
P.E.s = kx
2
P.E.s = strain energy in [ J ]
k = spring constant in [ N / m]
x = 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.
( K .E. + P.E.g + P.E.s )initial = ( K .E. + P.E.g + P.E.s ) final
K .E. = kinetic energy
P.E.g = gravitational potential energy
P.E.s = 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.