An elevator and its load have a combined mass of 1600 kg. Find the tension in the supporting cable when the elevator, originally moving downward at 12 m/s is brought to rest with constant acceleration in a distance of 42m.

T=w which means that T=mg. My book said that the answer is 1.8X10^4 N. But I can't get the answer. What am I doing wrong?

You must apply Newton's second Law:

F = M a,

where F is the force acting on a mass m and a is the acceleration of that mass.

Gravity acts on the elevator with a force in the downward direction of F_g = 1600 kg*9.81 m/s^2 = 15696 N

If the cable exerts a force T on the elevator in the upward direction of T, then the total force exerted on the elevator in the downward direction is:

F_{total} = 15696 N - T

The acceleration in the downward direction is then:

a = F_{total}/m = g - T/M

Now you ca deive from the given data that the lift is accelerating in the upward direction. You can write the acceleration in the downward direction as:

a = -1/2 (12 m/s)^2/(42m) = -1.714 m/s^2

So:

g - T/M = -1.714 m/s^2 -->

T = 1.8*10^4 N

(Note that it is a good practise to keep more significant figures in the intermediary steps and round it off at the end).

It seems like you're on the right track!

To find the tension in the supporting cable, let's break it down step by step:

1. Start by finding the force of gravity acting on the elevator and its load. This can be calculated using the formula Fg = mass * acceleration due to gravity. In this case, Fg = 1600 kg * 9.81 m/s^2 = 15696 N.

2. Next, consider that the cable exerts an upward force (tension) on the elevator. Let's call this T.

3. The total force exerted on the elevator in the downward direction is the difference between the force of gravity and the tension force: F_total = Fg - T.

4. Now, apply Newton's second law, which states that F = m * a. In this case, the total force is equal to the mass of the elevator multiplied by its acceleration: F_total = mass * acceleration.

5. Rearrange the equation to solve for acceleration: acceleration = F_total / mass.

6. Substitute the values: acceleration = (Fg - T) / mass.

7. Given in the problem is the fact that the elevator is brought to rest with constant acceleration over a distance of 42 m. So, we can determine the acceleration using the kinematic equation vf^2 = vi^2 + 2 * a * d.

vf = 0 (since it comes to rest)
vi = 12 m/s (initial velocity)
a = ? (acceleration)
d = 42 m (distance)

Substitute the values: 0^2 = 12^2 + 2 * a * 42.

Simplify and solve for a: a = (-12^2) / (2 * 42) = -1.714 m/s^2.

Note: The negative sign indicates that the elevator is accelerating in the upward direction.

8. Now, substitute the acceleration value into the equation from step 6: -1.714 m/s^2 = (Fg - T) / mass.

9. Rearrange the equation to solve for T: T = Fg - (mass * acceleration).

10. Plug in the given values: T = 15696 N - (1600 kg * -1.714 m/s^2).

11. Calculate T: T = 15696 N + 2745.6 N = 18441.6 N.

12. Round the answer to the appropriate number of significant figures: T ≈ 1.8 × 10^4 N.

So, the tension in the supporting cable is approximately 1.8 × 10^4 N, which matches the answer given in your book.

Based on the information provided, your calculation seems correct. The tension in the supporting cable is indeed 1.8*10^4 N. It appears that the book's answer is consistent with this result as well. Therefore, it is likely that there was an error in your calculations or a misunderstanding in the problem. Please double-check your work and make sure all the values used are accurate. If you are still having trouble, feel free to provide your step-by-step calculations, and I can help you identify where the mistake might have occurred.