Karen has a mass of 52.7 kg as she rides
the up escalator at Woodley Park Station of the Washington D.C. Metro. Karen rode a
distance of 57.5 m, the longest escalator in the free world. The acceleration of gravity is 9.8 m/s2 . How much work did the escalator do on
Karen if it has an inclination of 26.7�?
Answer in units of J
Work = Vertical Rise*Weight
= 57.5*sin26.7*M*g = ?
Well, Karen sure took quite the ride! So, to calculate the work done by the escalator on Karen, we need to use the formula:
Work = Force × Distance × cos(theta)
Where:
- Force is the gravitational force acting on Karen (mass × acceleration due to gravity).
- Distance is the distance Karen rode on the escalator.
- theta is the angle of inclination (26.7 degrees).
Let's plug in the values and calculate:
Force = 52.7 kg × 9.8 m/s²
Distance = 57.5 m
theta = 26.7 degrees
Work = (52.7 kg × 9.8 m/s²) × (57.5 m) × cos(26.7 degrees)
Calculating that out will give us the answer in units of Joules (J). Hold on, the clown calculator is crunching the numbers... And the answer is:
Work = 19602 J (approximately)
So, the escalator did about 19602 Joules of work on Karen! That's quite a workout without even moving a muscle!
To find the work done by the escalator on Karen, we need to calculate the gravitational potential energy change.
The formula for calculating gravitational potential energy is:
PE = m * g * h
Where:
PE represents the gravitational potential energy
m is the mass (in kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height or vertical distance (in m)
In this case, the vertical distance or height can be calculated using the inclined length and the angle of inclination.
h = distance * sin(angle)
Let's calculate the height first:
h = 57.5 m * sin(26.7 degrees)
h ≈ 57.5 m * 0.448
h ≈ 25.82 m
Now, we can calculate the gravitational potential energy change:
PE = m * g * h
PE = 52.7 kg * 9.8 m/s^2 * 25.82 m
PE ≈ 13,131.2 J
Therefore, the work done by the escalator on Karen is approximately 13,131.2 J.
To calculate the work done by the escalator on Karen, we need to determine the change in gravitational potential energy. The work done is given by the formula:
Work = Force × Distance × cos(theta)
Where:
Force = mass × acceleration due to gravity
Distance = distance traveled by Karen
theta = angle of inclination
First, let's calculate the force:
Force = mass × acceleration due to gravity
= 52.7 kg × 9.8 m/s^2
= 516.46 N
Next, let's calculate the distance traveled by Karen:
Distance = 57.5 m
Now, let's calculate the angle of inclination in radians:
theta = 26.7 degrees × (pi/180)
= 0.465 radians
Finally, we can calculate the work done by the escalator:
Work = Force × Distance × cos(theta)
= 516.46 N × 57.5 m × cos(0.465)
≈ 23139.544 J
Therefore, the work done by the escalator on Karen is approximately 23139.544 J (joules).