An elevator with a weight of 27.9 kN is given an upward acceleration of 1.46 m/s2 by a cable.
(a) Calculate the tension in the cable.
(b)What is the tension when the elevator is decelerating at the rate of 1.46 m/s2 but is still moving upward?
23 n
To solve this problem, we need to calculate the tension in the cable for both cases: when the elevator is accelerating upward and when it is decelerating but still moving upward.
(a) Calculating the tension when the elevator is accelerating upward:
When the elevator is accelerating upward, the net force acting on it is the vector sum of its weight (mg) and the tension force in the cable.
The formula for calculating the net force is:
Net force = mass * acceleration
Given:
Weight of the elevator (mg) = 27.9 kN = 27.9 * 1000 N = 27900 N (since 1 kN = 1000 N)
Acceleration (a) = 1.46 m/s²
Let's assume the mass of the elevator is represented by m.
The net force equation can be written as:
Tension - Weight of the elevator = mass * acceleration
T - 27900 = m * 1.46
We need one more equation to solve for both the tension and the mass. The equation for weight is:
Weight of the elevator = mass * gravitational acceleration
Weight of the elevator = m * 9.8 (taking gravitational acceleration as 9.8 m/s²)
Now we can substitute the weight equation into the net force equation:
T - m * 9.8 = m * 1.46
Simplifying the equation:
T = m * (1.46 + 9.8)
T = m * 11.26
(b) Calculating the tension when the elevator is decelerating but still moving upward:
When the elevator is decelerating but still moving upward, the net force acting on it is the vector sum of the tension force in the cable and the force due to deceleration.
The formula for calculating the net force is the same:
Net force = mass * acceleration
In this case, the acceleration is the deceleration rate of 1.46 m/s² but with a negative sign, as it is acting in the opposite direction. So the acceleration value is -1.46 m/s².
The net force equation can be written as:
Tension - Force due to deceleration (Weight of the elevator) = mass * acceleration
T - 27900 = m * (-1.46)
Again, substitute the weight equation into the net force equation:
T - m * 9.8 = m * (-1.46)
Simplifying the equation:
T = m * (-1.46 + 9.8)
T = m * 8.34
So, to calculate the tension in the cable in both cases, we need to know the mass of the elevator.