a 3200-lb elevator cab is supported by a cable in which the maximum safe tension is 4000 lb. (a)what is the greatest upward acceleration the elevator cab can have? (b) the greatest downward acceleration?
m=W/g
For upward motion
ma= T- mg
a=( T- mg)/m=(T-W)/m
For downward motion
ma=mg-T
a = (mg-T)/m=(W-T)/m
(a) Well, it seems like the elevator cab has been hitting the weights pretty hard. I mean, 3200 pounds? That's quite the heavy load! But fear not, we have a maximum safe tension of 4000 pounds. So the greatest upward acceleration the elevator cab can have would be when the tension in the cable equals the maximum safe tension.
To calculate this, we can use Newton's second law, which states that force equals mass times acceleration (F = ma). In this case, the force is the tension in the cable, which is 4000 pounds, and the mass is 3200 pounds. So we have:
4000 pounds = 3200 pounds * acceleration
Solving for acceleration, we find:
acceleration = 4000 pounds / 3200 pounds ≈ 1.25
Therefore, the greatest upward acceleration the elevator cab can have is approximately 1.25 times the acceleration due to gravity.
(b) Now, let's look at the greatest downward acceleration. Since the maximum safe tension is 4000 pounds, the tension in the cable needs to be less than or equal to that for it to be safe. So the greatest downward acceleration would be when the tension in the cable equals the maximum safe tension of 4000 pounds.
Using the same equation from before, we have:
4000 pounds = 3200 pounds * acceleration_down
Solving for acceleration_down, we find:
acceleration_down = 4000 pounds / 3200 pounds ≈ 1.25
Therefore, the greatest downward acceleration the elevator cab can have is approximately 1.25 times the acceleration due to gravity.
So, it looks like the greatest upward and downward accelerations the elevator cab can have are the same, approximately 1.25 times the acceleration due to gravity. It's all about keeping things in balance, just like a good circus act!
To determine the greatest upward and downward acceleration of the elevator cab, we need to analyze the tension forces acting on the cable.
First, let's consider the greatest upward acceleration:
(a) Greatest Upward Acceleration:
To determine the maximum safe upward acceleration, we need to find the net force acting on the elevator cab and divide it by its mass.
1. Calculate the weight of the elevator cab:
Weight = Mass * Acceleration due to gravity
Weight = 3200 lb * 32.2 ft/s^2 (approximate acceleration due to gravity)
Weight = 102,400 lb-ft/s^2
2. Subtract the maximum tension force from the weight to find the net force:
Net Force = Weight - Maximum Tension
Net Force = 102,400 lb-ft/s^2 - 4000 lb
Net Force = 98,400 lb-ft/s^2
3. Determine the net acceleration:
Net Acceleration = Net Force / Mass
Net Acceleration = 98,400 lb-ft/s^2 / 3200 lb
Net Acceleration ≈ 30.75 ft/s^2
Therefore, the greatest upward acceleration the elevator cab can have is approximately 30.75 ft/s^2.
Now, let's find the greatest downward acceleration:
(b) Greatest Downward Acceleration:
In this case, we need to consider the maximum safe tension acting in the opposite direction.
1. Calculate the maximum safe tension acting downward:
Maximum Safe Tension = Weight + Maximum Tension
Maximum Safe Tension = 102,400 lb-ft/s^2 + 4000 lb
Maximum Safe Tension = 106,400 lb-ft/s^2
2. Subtract the maximum safe tension from the weight to find the net force:
Net Force = Weight - Maximum Safe Tension
Net Force = 102,400 lb-ft/s^2 - 106,400 lb-ft/s^2
Net Force = -4000 lb-ft/s^2
3. Determine the net acceleration:
Net Acceleration = Net Force / Mass
Net Acceleration = -4000 lb-ft/s^2 / 3200 lb
Net Acceleration ≈ -1.25 ft/s^2
Therefore, the greatest downward acceleration the elevator cab can have is approximately -1.25 ft/s^2. The negative sign indicates a downward direction.
To find the greatest upward and downward accelerations of the elevator cab, we need to consider the forces acting on the elevator cab.
(a) Greatest upward acceleration:
To calculate the greatest upward acceleration, we need to consider the tension in the cable and the weight of the elevator cab.
The tension in the cable should be equal to or less than the maximum safe tension. So, Tension (T) ≤ 4000 lb.
The weight of the elevator cab can be calculated using the formula:
Weight (W) = mass × acceleration due to gravity.
Here, the mass of the elevator cab is given as 3200 lb.
The acceleration due to gravity is approximately 32.2 ft/s² (or 9.8 m/s²).
Now, substitute the values into the formula to find the weight:
W = 3200 lb × 32.2 ft/s².
Next, equate the tension in the cable to the sum of the weight and the tension required to accelerate the elevator cab upward (T + W).
T + W ≤ 4000 lb.
Rearrange the equation to isolate the tension in the cable by subtracting the weight from both sides:
T ≤ 4000 lb - W.
Finally, substitute the value of W and solve for T:
T ≤ 4000 lb - (3200 lb × 32.2 ft/s²).
Once you have the maximum tension in the cable (T), divide it by the mass of the elevator cab to find the greatest upward acceleration:
a = T / mass.
(b) Greatest downward acceleration:
The calculation for the greatest downward acceleration is similar to the previous one. However, in this case, the tension in the cable will be the difference between the weight of the elevator cab and the maximum safe tension:
T ≤ W - 4000 lb.
Substitute the values and solve for T:
T ≤ (3200 lb × 32.2 ft/s²) - 4000 lb.
Then, divide the maximum tension by the mass of the elevator cab to find the greatest downward acceleration:
a = T / mass.
Remember, all calculations should be performed using consistent units. Also, make sure to account for the gravitational acceleration specific to your unit system.