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Gravity and Gravitational Potential Energy
In free fall, the only force acting on the mass is the force
of gravity. So the net force acting on the object is Fg
which is equal to mg. The displacement is the change in
height, Δh. So the work done by gravity is
Wg = Fgd
or
Wg = mgΔh
The force of gravity is a force that can do either positive
or negative work.
For example, a mass is falling vertically:
Fg
d = Δh
The force is downward and the displacement is
downward, so the work done by gravity, Wg, is positive.
If you throw a ball up in the air:
d = Δh
Fg
The force is downward and the displacement is upward,
so the work done by gravity, Wg, is negative.
Example 1) A ball is thrown up into the air from a
height of 1.2 m. The ball reaches a maximum height of
4.8 m before falling back down.
a. Using the principles of work and energy, determine
the speed with which the ball leaves the person’s
hand.
W = mgΔh
and
W = KE2 – KE1
At the maximum height, the velocity is 0, so KE2 = 0
b. Determine the speed of the ball after it falls back
down to a height of 3.5 m using the principles of
work and energy.
W = mgΔh
and
W = KE2 – KE1
At the maximum height, the velocity is 0, KE__ = 0
Gravitational Potential Energy
Gravitational potential energy is the potential energy an
object has because of its location in a gravitational field.
Objects at higher altitudes have greater gravitational
potential energy than objects at lower altitudes.
An example of gravitational potential energy would be a
piledriver. It contains a large hammer that is raised to a
height and then dropped. The potential energy is
transformed into sufficient energy to drive the pile into the
ground. The higher it is raised, the greater the potential
energy! The work done by the piledriver is the change is
potential energy.
Wg = PE2 – PE1
Work done by gravity is Wg = mgΔh or written differently:
Wg = mgh2 – mgh1
So Gravitational Potential Energy, PE, is the product of
the mass, acceleration due to gravity, and the height.
PE = mgh
Where m is the scalar quantity, mass, in kg
g is the acceleration of gravity,
h is the scalar quantity height, in meters.
PE is the scalar quantity, Potential Energy in Joules (J)
Remember, Work is the change in gravitational potential
energy,
Wg = ΔPE = PE2 – PE1 = mgΔh
Example 2) A person has a mass of 65 kg and is
rollerblading down a hill. At one point, the person is a
vertical distance of 25 m above the bottom of the hill.
Sometime later, the person is a vertical distance of 12 m
above the bottom of the hill.
Remember: the vertical components are independent
from the horizontal components, so when determining the
work done by gravity, we can consider the vertical
components only.
a. Determine the initial gravitational potential energy of
the person.
b. Determine the final gravitational potential energy of
the person.
c. Determine the change in gravitational potential
energy of the person.
d. Determine the work done by gravity.
Gravity and Gravitational Potential Energy
In free fall, the only force acting on the mass is the force of gravity. So the net force acting on
the object is Fg which is equal to mg. The displacement is the change in height, Δh. So the work
done by gravity is
Wg = Fgd
or
Wg = mgΔh
The force of gravity is a force that can do either positive or negative work.
For example, a mass is falling vertically:
F
g
d = Δh
The force is downward and the displacement is downward, so the work done by gravity, Wg, is
positive.
If you throw a ball up in the air:
d = Δh
F
g
The force is downward and the displacement is downward, so the work done by gravity, Wg, is
positive.
Example 1) A ball is thrown up into the air from a height of 1.2 m. The ball reaches a maximum
height of 4.8 m before falling back down.
a. Using the principles of work and energy, determine the speed with which the ball
leaves the person’s hand.
W = mgΔh
and
W = KE2 – KE1
At the maximum height, the velocity is 0, so KE2 = 0
b. Determine the speed of the ball after it falls back down to a height of 3.5 m using the
principles of work and energy.
W = mgΔh
and
W = KE2 – KE1
At the maximum height, the velocity is 0, KE__ = 0
Gravitational Potential Energy
Gravitational potential energy is the potential energy an object has because of its location in a
gravitational field. Objects ay higher altitudes have greater gravitational potential energy than
objects at lower altitudes.
An example of gravitational potential energy would be a piledriver. It contains a large hammer
that is raised to a height and then dropped. The potential energy is transformed into sufficient
energy to drive the pile into the ground. The higher it is raised, the greater the potential energy!
The work done by the piledriver is the change is potential energy.
Wg = PE2 – PE1
Work done by gravity is Wg = mgΔh or written differently:
Wg = mgh2 – mgh1
So Gravitational Potential Energy, PE, is the product of the mass, acceleration due to gravity,
and the height.
PE = mgh
Where m is the scalar quantity, mass, in kg
g is the acceleration of gravity,
h is the scalar quantity height, in meters.
PE is the scalar quantity, potential Energy in Joules (J)
Remember, Work is the change in gravitational potential energy,
Wg = ΔPE = PE2 – PE1 = mgΔh
Example 2) A person has a mass of 65 kg and is rollerblading down a hill. At one point, the
person is a vertical distance of 25 m above the bottom of the hill. Sometime later, the person is
a vertical distance of 12 m above the bottom of the hill.
Remember: the vertical components are independent from the horizontal components, so when
determining the work done by gravity, we can consider the vertical components only.
a.
b.
c.
d.
Determine the initial gravitational potential energy of the person.
Determine the final gravitational potential energy of the person.
Determine the change in gravitational potential energy of the person.
Determine the work done by gravity
Learning Activity 4.3 – Gravity, Work, and Potential Gravitational Energy
1. A stone has a mass of 50 g and is thrown upwards at 22 m/s at a point 5 m above the
ground.
a. What is the work done by gravity during the time the stone rises to a new height of
19 m above the ground.
b. Calculate the velocity of the stone when it is 19 m above the ground.
c. What is the speed of the stone as it crashes into the ground.
2. A piledriver of mass 475 kg falls a distance of 6.25 m before it strikes a pile. Determine
the change in gravitational potential energy of the piledriver.
3. A person has a mass of 75 kg and walks up the stairs from the first floor to the fourth
floor. The vertical distance between each floor is 3m
a. Determine the gravitational potential energy of the person when he is standing on
the landing for the third floor.
b. Determine the work done by gravity if a 45 kg package is dropped from the fourth
floor to the first floor.
4. An arrow is shot up in the air from the top of a parapet that is 35 meters in the air. The
arrow reaches a maximum height of 62 m. What is the vertical speed of the arrow as it is
shot into the air?