Karen has a mass of 48.2 kg as she rides
the up escalator at Woodley Park Station of
the Washington D.C. Metro. Karen rode a
distance of 65.7 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 32.9�?
Answer in units of J
Wt. = 48.2kg * 9.8N/kg = 452.8N.
Fp = 452.8*sin32.9 = 245.95N = Force
parallel to plane.
W=Fp*d = 245.95 * 65.7 = 16159 Joules.
To determine the work done by the escalator on Karen, we can use the formula:
Work = force × distance × cos(theta)
Where:
- force is the force exerted by Karen's weight,
- distance is the distance Karen rode on the escalator, and
- theta is the angle of inclination of the escalator.
First, let's calculate the force exerted by Karen's weight:
Force = mass × gravity
Given that the mass of Karen is 48.2 kg and the acceleration due to gravity is 9.8 m/s^2, we can substitute these values into the equation:
Force = 48.2 kg × 9.8 m/s^2 = 472.36 N
Now we can calculate the work done by the escalator:
Work = 472.36 N × 65.7 m × cos(32.9°)
Make sure to convert the angle from degrees to radians before calculating the cosine:
θ (in radians) = θ (in degrees) × (π/180)
θ (in radians) = 32.9° × (π/180) ≈ 0.5735 rad
Work = 472.36 N × 65.7 m × cos(0.5735 rad)
Using a calculator, we can evaluate the cosine:
Work ≈ 472.36 N × 65.7 m × 0.8470
Finally, we can calculate the work:
Work ≈ 26814.6572 J
Therefore, the work done by the escalator on Karen is approximately 26814.6572 Joules (J).
To calculate the work done by the escalator, we need to find the force exerted by the escalator on Karen and the distance she traveled.
First, let's find the gravitational force acting on Karen. The formula for gravitational force is given by:
F_gravity = mass * acceleration due to gravity
F_gravity = 48.2 kg * 9.8 m/s^2
F_gravity = 472.36 N
Next, let's find the vertical component of the force due to the escalator's inclination. This force can be calculated using:
F_inclination = mass * acceleration due to gravity * sin(theta)
Where:
mass = 48.2 kg (Karen's mass)
acceleration due to gravity = 9.8 m/s^2
theta = 32.9 degrees (convert to radians)
theta_radians = 32.9 degrees * π / 180
theta_radians = 0.57 radians
F_inclination = 48.2 kg * 9.8 m/s^2 * sin(0.57 radians)
F_inclination ≈ 277.32 N
Note: sin(0.57 radians) is not a whole number, so we are using the approximate value.
The work done by the escalator is the dot product of the force exerted by the escalator and the distance traveled:
Work = F_inclination * distance
Work = 277.32 N * 65.7 m
Work ≈ 18,214.10 J
Therefore, the work done by the escalator on Karen is approximately 18,214.10 Joules (J).
m g h
h = 65.7 sin 32.9