You are moving into an apartment and take the elevator to the 6th floor. Suppose your weight is 750 N and that of your belongings is 990 N. (a) Determine the work done by the elevator in lifting you and your belongings up to the 6th floor (15.2 m) at a constant velocity. (b) How much work does the elevator do on you alone (without belongings) on the downward trip, which is also made at a constant velocity?
a. work=totalmass*g*height
b. work=yourmass*g*height
c.-mgh
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(a) To determine the work done by the elevator in lifting you and your belongings up to the 6th floor at a constant velocity, we can use the formula:
Work = Force x Distance x Cos(θ)
Where:
Force = Weight of you and your belongings
Distance = Height to the 6th floor
θ = Angle between the applied force and the direction of motion (which is 0° in this case since the elevator is moving vertically)
Given:
Weight of you and your belongings = 750 N + 990 N = 1740 N
Distance = 15.2 m
θ = 0°
Plugging in the values into the formula, we get:
Work = 1740 N x 15.2 m x Cos(0°)
Cos(0°) = 1, since Cos(0°) = 1
Therefore, Work = 1740 N x 15.2 m x 1 = 26,448 Joules
So, the work done by the elevator in lifting you and your belongings up to the 6th floor at a constant velocity is 26,448 Joules.
(b) To determine the work done by the elevator on you alone (without belongings) on the downward trip, we need to consider the change in potential energy.
The work done by gravity while the elevator descends is equal to the negative change in potential energy. Since potential energy is given by P.E. = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height, the work done by gravity is given by the formula:
Work = -(Change in potential energy)
For you alone, we need to subtract the weight of your belongings from your total weight.
Given:
Weight of you = 750 N
Change in potential energy = mgΔh, where Δh is the change in height (15.2 m) and g = 9.8 m/s²
Plugging in the values into the formula, we get:
Work = -(750 N x 9.8 m/s² x 15.2 m)
Work = -112,560 Joules
Therefore, the work done by the elevator on you alone (without belongings) on the downward trip is -112,560 Joules. The negative sign indicates that work is done on you to decrease your potential energy.
To determine the work done in each scenario, we can use the formula for work:
Work (W) = Force (F) × Displacement (d) × cos(θ),
where:
- W is the work done,
- F is the force applied,
- d is the displacement, and
- θ is the angle between the force and displacement vectors.
(a) To find the work done by the elevator while lifting you and your belongings up to the 6th floor at a constant velocity:
1. Calculate the total force: Since you and your belongings are lifted together, add the weights of both, which gives a total force of:
F_total = Your weight + Belongings' weight = 750 N + 990 N = 1740 N.
2. Determine the displacement: The 6th floor is 15.2 m above the starting point; thus, the displacement is:
d = 15.2 m.
3. Calculate the angle between the force and displacement vectors: Since the elevator is moving vertically, the angle between them is 0°. Therefore, the cos(θ) is equal to 1.
4. Substitute the values into the work formula:
W_total = F_total × d × cos(θ)
= 1740 N × 15.2 m × cos(0°)
= 26448 Joules.
Therefore, the work done by the elevator in lifting you and your belongings to the 6th floor is 26448 Joules.
(b) To find the work done by the elevator on you alone (without belongings) on the downward trip (assuming a constant velocity):
1. Calculate the force: Since only your weight is taken into account this time, the force is:
F_you = Your weight = 750 N.
2. Determine the displacement: The downward trip from the 6th floor to the starting point has the same distance as before:
d = 15.2 m.
3. Calculate the angle between the force and displacement vectors: As the elevator moves vertically downwards, the angle between them is 180°. In this case, the cos(θ) is equal to -1.
4. Substitute the values into the work formula:
W_you = F_you × d × cos(θ)
= 750 N × 15.2 m × cos(180°)
= -11400 Joules (Note: The negative sign represents the opposite direction of the displacement).
Therefore, the work done by the elevator on you alone, during the downward trip, is -11400 Joules.