the floors in a typical house are separated by a vertical distance of approximately 8.0 ft. a teenager (1.0x 10^2 lb) climbs the stairs between floors at a steady speed. find the power necessary to accomplish this if the stairs are climbed in (a) 10.0 s and (b) 2.0 s. express the answers in units of horsepower.
a. h.p.=F*d/550t=100*8 / 550*10=0.1455.
b. h.p. = 100*8 / 550*2 = 0.7273.
To calculate the power necessary to climb the stairs, we can use the formula:
Power = Work / Time
First, let's calculate the work done to climb the stairs.
Work = Force × Distance
Given:
Vertical distance = 8.0 ft
Weight of teenager = 1.0 × 10^2 lb
(a) When the stairs are climbed in 10.0 s:
Force = Weight = 1.0 × 10^2 lb
Distance = 8.0 ft
Work = 1.0 × 10^2 lb × 8.0 ft
Now, let's convert the units to horsepower.
1 horsepower (hp) is equal to 550 ft-lb/s.
To convert the work to horsepower, we divide it by the time taken in seconds and then divide by 550 ft-lb/s.
Power (in hp) = (Work / Time) / 550 ft-lb/s
(b) When the stairs are climbed in 2.0 s:
The steps are the same as in part (a), but with a different time.
Now, let's calculate the power using the same formula:
Power = (Work / Time) / 550 ft-lb/s
I will calculate the power for both cases and provide the answers as required in units of horsepower.
To find the power necessary to climb the stairs, we need to calculate the work done in lifting the teenager through a vertical distance of approximately 8.0 ft. The equation for work done is:
Work = Force × Distance × Cos(θ)
In this case, the force is the weight of the teenager, the distance is the height between the floors, and θ is the angle between the force and the direction of motion (which is 0° since the teenager is climbing straight up). The weight force can be calculated using the equation:
Weight = mass × acceleration due to gravity
The acceleration due to gravity is approximately 32.2 ft/s². Therefore, the weight force of the teenager can be found by multiplying their mass by the acceleration due to gravity.
Now, let's calculate the power required using the formula:
Power = Work / Time
For part (a), where the stairs are climbed in 10.0 s, we can find the power required using the above equations. First, we'll calculate the weight force:
Weight = (1.0 × 10^2 lb) × (32.2 ft/s²)
Next, we'll calculate the work done:
Work = Weight × Distance
Work = (Weight) × (8.0 ft)
Finally, we'll use the formula for power:
Power = Work / Time
Power = (Work) / (10.0 s)
To convert the final result into horsepower, we'll use the conversion factor:
1 horsepower (hp) = 550 ft⋅lbf/s
Now, repeat the same calculations for part (b), where the stairs are climbed in 2.0 s.
Let's go through the calculations step by step:
Part (a):
1. Calculate weight:
Weight = (1.0 × 10^2 lb) × (32.2 ft/s²)
Weight = (320 lb) × (32.2 ft/s²)
2. Calculate work:
Work = Weight × Distance
Work = (Weight) × (8.0 ft)
3. Calculate power:
Power = Work / Time
Power = (Work) / (10.0 s)
4. Convert to horsepower:
Power (in horsepower) = Power / 550 ft⋅lbf/s
Part (b):
1. Calculate weight:
Weight = (1.0 × 10^2 lb) × (32.2 ft/s²)
Weight = (320 lb) × (32.2 ft/s²)
2. Calculate work:
Work = Weight × Distance
Work = (Weight) × (8.0 ft)
3. Calculate power:
Power = Work / Time
Power = (Work) / (2.0 s)
4. Convert to horsepower:
Power (in horsepower) = Power / 550 ft⋅lbf/s
By following these steps, you can find the power necessary to climb the stairs in both scenarios and express the answers in units of horsepower.