An elevator and its load have a combined mass of 1550 kg. Find the tension in the supporting cable in Newtons when the elevator , originally moving downward at 11 m/s, is brought to rest with constant acceleration in a distance of 43 m.
Tension= mg +- ma where a is the acceleration. + when the elevator is going up (it adds to tension), and - going down, it reduces tension when falling.
a= change velocity/time (rest means zero velocity)
To find the tension in the supporting cable, we first need to find the net force acting on the elevator.
The net force can be calculated using Newton's second law of motion:
Net Force = Mass * Acceleration
In this case, the elevator is brought to rest with constant acceleration, so its final velocity is 0 m/s. The initial velocity is given as 11 m/s.
Using the equation for constant acceleration:
Final velocity^2 = Initial velocity^2 + 2 * acceleration * distance
Substituting the known values:
0^2 = 11^2 + 2 * acceleration * 43
Rearranging the equation to solve for acceleration:
Acceleration = (0^2 - 11^2) / (2 * 43)
Acceleration = -121 / 86
Acceleration = -1.407 m/s^2 (negative because the elevator is decelerating)
Now, we can calculate the net force:
Net Force = Mass * Acceleration
Net Force = 1550 kg * -1.407 m/s^2
Net Force = -2179.35 N
Since we are interested in the tension in the cable, we need to consider the direction of the force. The tension in the cable acts upward, opposite to the force of gravity.
Therefore, the tension in the cable will be equal to the magnitude of the net force plus the force of gravity:
Tension = |Net Force| + Force of Gravity
The force of gravity can be calculated using the equation:
Force of Gravity = Mass * Gravity
Assuming a standard gravity of 9.8 m/s^2:
Force of Gravity = 1550 kg * 9.8 m/s^2
Force of Gravity = 15190 N
Tension = |-2179.35 N| + 15190 N
Tension = 17369.35 N
Therefore, the tension in the supporting cable is approximately 17369.35 Newtons.