In which of these situations, is mechanical energy being conserved? (Neglect, air resistance, friction, and breaking) Check all
1 answer:
1) Mechanical energy is conserved in all the situations listed
2) True
3) The energy that an object has stored due to its position or shape is called potential energy
4) The potential energy of the block is 490 J
5) The potential energy of the elevator is 750,000 J
Explanation:
1)
The mechanical energy of an object is the sum of its kinetic energy (KE) and its potential energy (PE):
Where
KE is the energy due to the motion of the object
PE is the energy due to the position of the object (it can be either gravitational potential energy or elastic potential energy)
In absence of non-conservative forces, such as friction or air resistance, the mechanical energy is always conserved. Therefore, the mechanical energy is conserved in all the situations listed here:
Child on a swing --> there is a continuous conversion between gravitational potential energy and kinetic energy
Pendulum --> there is a continuous conversion between gravitational potential energy and kinetic energy
Bow and Arrow --> there is a conversion between elastic potential energy of the bow and kinetic energy of the arrow
Roller Coaster --> there is a continuous conversion between gravitational potential energy and kinetic energy
2)
The potential energy of an object is given by
where
m is its mass
g is the acceleration due to gravity
h is the height of the object relative to the ground
While the kinetic energy is given by
where
v is the speed of the object
As an object falls to the ground, its height h decreases, therefore the potential energy PE decreases as well. However, the speed of the object, v, increases during the fall, and therefore the kinetic energy KE increases. This means that potential energy is converted into kinetic energy.
3)
Potential energy is the energy possessed by an object due to its position. It can be of two types:
- Gravitational potential energy: it is the potential energy due to the position of an object in a gravitational field. It is calculated as
, as shown in part 2)
- Elastic potential energy: it is the potential energy stored in an elastic object when it is stretched or compressed. It is calculated as
, where k is the spring constant of the elastic object and x is the stretching/compression of the object relative to its equilibrium position.
4)
The potential energy stored in an object held above the ground is given by
where
m is the mass of the object
g is the acceleration of gravity
h is the height of the object relative to the ground
For the object in this problem, we have
m = 10 kg
h = 5 m
Substituting, we find
5)
As before, the potential energy of the elevator is given by
where m is its mass and h is its height above the ground.
Here we don't have the mass of the elevator. However, we know its weight:
But we also know that the weight of an object is equal to the product between its mass and the acceleration of gravity:
So we can rewrite the potential energy as
and the height of the elevator is
h = 500 m
Therefore, its potential energy is
Learn more about potential energy:
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(a) 10241 W
In this situation, the car is moving at constant speed: this means that its acceleration along the direction parallel to the slope is zero, and so the net force along this direction is also zero.
The equation of the forces along the parallel direction is:
where
F is the force applied to pull the car
m = 950 kg is the mass of the car
is the acceleration of gravity
is the angle of the incline
Solving for F,
Now we know that the car is moving at constant velocity of
v = 2.20 m/s
So we can find the power done by the motor during the constant speed phase as
(b) 10624 W
The maximum power is provided during the phase of acceleration, because during this phase the force applied is maximum. The acceleration of the car can be found with the equation
where
v = 2.20 m/s is the final velocity
a is the acceleration
u = 0 is the initial velocity
t = 12.0 s is the time
Solving for a,
So now the equation of the forces along the direction parallel to the incline is
And solving for F, we find the maximum force applied by the motor:
The maximum power will be applied when the velocity is maximum, v = 2.20 m/s, and so it is:
(c)
Due to the law of conservation of energy, the total energy transferred out of the motor by work must be equal to the gravitational potential energy gained by the car.
The change in potential energy of the car is:
where
m = 950 kg is the mass
is the acceleration of gravity
is the change in height, which is
where L = 1250 m is the total distance covered.
Substituting, we find the energy transferred: