A 55 kg engineer leaves her office on the 33rd floor of a skyscraper and takes an elevator to the 59th floor. Later she descends to street level. If the engineer takes her office as the zero for potential energy and if the distance from one floor to the next is 3.5 m, what is the engineer’s potential energy (a) in her office, (b) on the 59th floor, and (c) at street level?
To find the engineer's potential energy at different locations, we need to first calculate the height of each location.
Given:
Mass (m) = 55 kg
Height between each floor (h) = 3.5 m
(a) Potential energy in her office:
Since the office is on the 33rd floor, we need to calculate the height of 33 floors below the office.
Height (h) = 33 * 3.5 m
(b) Potential energy on the 59th floor:
Since the engineer takes an elevator to the 59th floor, we need to calculate the height from the office to the 59th floor.
Height (h) = 26 * 3.5 m
(c) Potential energy at street level:
The engineer descends from the 59th floor to street level, which is a total of 59 floors.
Height (h) = 59 * 3.5 m
Now, we can calculate the potential energy using the formula:
Potential Energy (PE) = mass (m) * gravity (g) * height (h)
where,
mass (m) = 55 kg
gravity (g) = 9.8 m/s² (acceleration due to gravity)
Let's calculate the potential energy at each location:
(a) Potential energy in her office:
PE_office = m * g * height_office
(b) Potential energy on the 59th floor:
PE_59th_floor = m * g * height_59th_floor
(c) Potential energy at street level:
PE_street_level = m * g * height_street_level
Substituting known values:
PE_office = 55 kg * 9.8 m/s² * (33 * 3.5 m)
PE_59th_floor = 55 kg * 9.8 m/s² * (26 * 3.5 m)
PE_street_level = 55 kg * 9.8 m/s² * (59 * 3.5 m)
Now, let's calculate the values:
(a) Potential energy in her office:
PE_office = 55 kg * 9.8 m/s² * (33 * 3.5 m)
(b) Potential energy on the 59th floor:
PE_59th_floor = 55 kg * 9.8 m/s² * (26 * 3.5 m)
(c) Potential energy at street level:
PE_street_level = 55 kg * 9.8 m/s² * (59 * 3.5 m)
Calculating each value:
(a) PE_office = 55 kg * 9.8 m/s² * (33 * 3.5 m)
(b) PE_59th_floor = 55 kg * 9.8 m/s² * (26 * 3.5 m)
(c) PE_street_level = 55 kg * 9.8 m/s² * (59 * 3.5 m)
The potential energy at each location is:
(a) PE_office = 60945 J
(b) PE_59th_floor = 48986 J
(c) PE_street_level = 107673 J
Therefore,
(a) The engineer's potential energy in her office is 60945 J.
(b) The engineer's potential energy on the 59th floor is 48986 J.
(c) The engineer's potential energy at street level is 107673 J.
To calculate the engineer's potential energy at different locations, we need to determine the height (h) at each location and use the formula for potential energy:
Potential Energy (PE) = mass (m) × gravity (g) × height (h)
Given information:
- Mass (m) of the engineer = 55 kg
- Distance between each floor = 3.5 m
To calculate the height at each location:
- The engineer is leaving her office on the 33rd floor and going to the 59th floor, so the total number of floors she goes up is 59 - 33 = 26 floors.
- Since each floor has a height of 3.5 m, the total height she ascends is 26 × 3.5 m.
Now we can calculate the potential energy at each location:
(a) Potential energy in her office (zero potential energy): Since potential energy is defined in relation to a reference point, the engineer's potential energy in her office is zero.
(b) Potential energy on the 59th floor:
- Height (h) on the 59th floor = total number of floors ascended × height between each floor
- h = 26 × 3.5 m
- Plug in the values into the formula: PE = mass (m) × gravity (g) × height (h)
(c) Potential energy at street level:
- In this case, it's important to consider the direction of potential energy. Since the engineer is descending from a higher location to street level, the potential energy will be negative.
- Height (h) at street level = total number of floors ascended × height between each floor (negative value)
- h = -26 × 3.5 m
- Plug in the values into the formula: PE = mass (m) × gravity (g) × height (h)
gravity (g) is approximately 9.8 m/s², which is the standard gravity on Earth.
Using these calculations, you can find the engineer's potential energy at each location.
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mass m = 55 kg
Distance from one floor to the next h =3.5 m
(a). Potential energy in her office P= mg * 0 = 0 J
Since her office as the zero potential energy
(b). Potential energy on the 59 th floor = mg ( 59 th floor -33 floor )
= mg * 26 floors
= mg * 26 h
= 55*9.8 * 26 * 3.5
= 49049 J
(c). Potential energy at strret level = mg ( 0 thfloor - 33 floor )
= mg * -33 floors
= mg * -33h
= 55*9.8 * -33 * 3.5
= - 62254.5J