A fully loaded elevator weighing 2000 lbs accelerates upward from rest, attaining its rated velocity of 20 fps upon reaching a height of 25 ft. Solve for the tension on the cable lifting the elevator. What is the minimum required power of the lifting motor if 10% is allowed for friction?
I tried to solve this. I was not able to solve it. I don't know where I will start.
To solve for the tension on the cable lifting the elevator, we can use Newton's second law of motion, which states that the force applied to an object is equal to its mass times its acceleration.
First, let's calculate the acceleration of the elevator. We are given that the elevator starts from rest and reaches a velocity of 20 fps. We can use the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement. Since the elevator starts from rest, its initial velocity (u) is 0.
(20 fps)^2 = (0 fps)^2 + 2 * a * 25 ft
400 fps^2 = 50 ft * a
a = 400 fps^2 / 50 ft
a = 8 fps^2
Next, we can calculate the mass (m) of the elevator using its weight (W). The weight of the elevator is given as 2000 lbs. We can use the formula:
W = m * g
where g is the acceleration due to gravity, approximately 32.2 ft/s^2.
2000 lbs = m * 32.2 ft/s^2
m = 2000 lbs / 32.2 ft/s^2
m ≈ 62.11 slugs (slug is the unit of mass in English system)
Now we can calculate the tension (T) on the cable using Newton's second law:
T = m * a
T = 62.11 slugs * 8 fps^2
T ≈ 496.88 lb or lb-ft/s^2
To calculate the minimum required power of the lifting motor, we need to take into account the friction. We are given that 10% of the power is used for friction, and the minimum required power is the power needed to overcome the friction and lift the elevator.
Let the total power required be P.
Power = T * v
P = T * 20 fps
Next, we need to consider the power lost due to friction, which is 10% of the total power.
Power lost due to friction = 0.10 * P
Power lost due to friction = 0.10 * (T * 20 fps)
Equating the power lost due to friction to the tension multiplied by the velocity gives:
0.10 * (T * 20 fps) = T * 20 fps
0.10 * T * 20 fps = T * 20 fps
0.10 * T = T
Now we can solve for T to find the tension:
0.10 * T = T
0.10 = 1
This equation is not possible as 0.10 is not equal to 1.
Therefore, there seems to be an error in the given problem or data provided.
To solve this problem, we can use Newton's second law of motion and the work-energy principle. Let's break down the steps to find the tension on the cable and the minimum required power of the lifting motor:
Step 1: Finding the net force on the elevator
First, we need to calculate the net force acting on the elevator. Since the elevator is accelerating upward, the net force can be found using Newton's second law of motion:
Net force = mass × acceleration
Given:
Mass of the elevator = 2000 lbs (converted to mass in pounds by dividing by the acceleration due to gravity, approximately 32.2 ft/s^2)
The elevator is fully loaded, so its mass will be the same as the weight (force due to gravity) because mass = weight ÷ acceleration due to gravity.
Therefore:
Mass of the elevator = 2000 lbs ÷ 32.2 ft/s^2
Acceleration of the elevator can be calculated using the following formula:
Acceleration = change in velocity ÷ time
Given:
Final velocity (v) = 20 ft/s
Time taken (t) = ?
Since the elevator starts from rest (zero velocity), we can find the time taken by using the equation of motion:
v = u + at
where:
u = initial velocity (which is 0)
a = acceleration
Solving for time (t):
20 ft/s = 0 ft/s + a(t)
a(t) = 20 ft/s
t = 20 ft/s ÷ (acceleration)
Step 2: Finding the acceleration
To find the acceleration, we can use the kinematic equation:
s = ut + (1/2)at^2
Given:
Distance (s) = 25 ft
Initial velocity (u) = 0 ft/s
Time taken (t) = found in Step 1
Substituting the given values:
25 ft = 0 ft/s × t + (1/2) a(t^2)
25 ft = (1/2) a(t^2)
Solving for acceleration (a):
a = (2 × 25 ft) ÷ (t^2)
Step 3: Finding the friction force
Next, we need to determine the friction force. Since the problem states that 10% of the power is lost to friction, we can use the following equation:
Friction force = 0.10 × Net force
Step 4: Finding the tension on the cable
To find the tension on the cable, we need to consider all the forces acting on the elevator. The forces involved are the gravitational force (weight), tension force, and friction force. The net force acting on the elevator is the sum of these forces.
Net force = Tension force - Weight - Friction force
Since the elevator is moving upward, the tension force should be greater than the weight and the friction force:
Net force = Tension force - Weight - Friction force > 0
Solving for the tension force:
Tension force > Weight + Friction force
Step 5: Finding the minimum required power
The power required by the lifting motor can be calculated using the following formula:
Power = Net force × velocity
Given:
Net force = found in Step 4
Velocity (v) = 20 ft/s
Substituting the values, we can find the minimum required power.
That's how you can approach this problem step by step. Let me know if you need assistance with any specific calculation.