A skier of mass 76 kg is pulled up a slope by a motor-driven cable.
(a) How much work is required to pull him 40 m up a 30° slope (assumed frictionless) at a constant speed of 2.5 m/s?
J
I know that the answer to the first problem is 14896Joules, but I do not know how to convert to hp
(b) What power must a motor have to perform this task?
(a) The energy required to do the lifting is M g 40 sin 30 = 14,900 J
(b) Divide the work required by the time. To pull 40 m at a rate of 2.5 m/s will require a time of 40/2.5 = 16 seconds.
The power required will be in Watts.
746 W = 1 Horsepower, in case you want to convert to HP
To find the work required to pull the skier up the slope, we can use the formula:
Work = Force x Distance x cos(angle)
First, let's find the force. The gravitational force pulling the skier down the slope can be calculated using their mass (m) and the acceleration due to gravity (g).
Force = mass x acceleration due to gravity
Force = 76 kg x 9.8 m/s^2
Force = 744.8 N
Now, let's find the distance the skier is being pulled up the slope (d) and convert it to meters:
Distance = 40 m
Next, let's calculate the angle between the slope and the horizontal (θ) in radians:
θ = 30°
θ = 30° x π/180
θ = 0.5236 radians
Now, let's calculate the work:
Work = Force x Distance x cos(angle)
Work = 744.8 N x 40 m x cos(0.5236)
Work = 14896 J
Therefore, the work required to pull the skier up the slope is 14896 Joules (J).
Now, let's move on to part (b) to calculate the power.
Power is defined as the rate at which work is done. It can be calculated using the formula:
Power = Work / Time
We're given that the skier is moving at a constant speed of 2.5 m/s. To find time (t), we can use the formula:
Time = Distance / Speed
Time = 40 m / 2.5 m/s
Time = 16 s
Now, let's calculate the power:
Power = Work / Time
Power = 14896 J / 16 s
Power = 931 W
To convert the power from watts (W) to horsepower (hp), we can use the conversion factor:
1 hp = 745.7 W
Let's calculate the power in horsepower:
Power (hp) = Power (W) / 745.7
Power (hp) = 931 W / 745.7
Power (hp) ≈ 1.25 hp
Therefore, the power required for the motor to perform this task is approximately 1.25 horsepower.
To solve this problem, we first calculate the work done to pull the skier up the slope and then convert it to horsepower (hp).
(a) Calculating work done:
The work done (W) is given by the formula: W = force x distance x cos(angle).
In this case, the force required to pull the skier up the slope is equal to the component of the skier's weight parallel to the slope. We can calculate it using the formula: force = mass x acceleration.
The acceleration along the slope can be found using the formula: acceleration = g x sin(angle), where g is the acceleration due to gravity (approximately 9.8 m/s²).
Substituting the given values:
mass = 76 kg
distance = 40 m
angle = 30°
First, we calculate the acceleration:
acceleration = g x sin(angle) = 9.8 m/s² x sin(30°) ≈ 4.9 m/s²
Next, we calculate the force:
force = mass x acceleration = 76 kg x 4.9 m/s² ≈ 372.4 N
Finally, we calculate the work done:
W = force x distance x cos(angle) = 372.4 N x 40 m x cos(30°)
Using a calculator or math software, we find:
W ≈ 14896 J
So, the work required to pull the skier up the slope is approximately 14896 Joules.
(b) Converting work to power:
Power (P) is defined as the rate at which work is done, given by the formula: P = work/time.
In this case, we are given the speed (constant speed of 2.5 m/s), and we can use it to calculate the time (t) required to complete the task.
Given:
distance = 40 m
speed = 2.5 m/s
Using the formula: time = distance/speed, we can find:
time = 40 m / 2.5 m/s = 16 s
Now, we calculate the power:
P = work/time = 14896 J / 16 s
To convert to horsepower (hp), we need to know that 1 hp is equivalent to 745.7 watts (W).
P_hp = P / 745.7
Using a calculator or math software, we find:
P_hp ≈ (14896 J / 16 s) / 745.7 ≈ 1.58 hp
Therefore, the motor must have approximately 1.58 horsepower to perform this task.