A large spring is placed at the bottom of an elevator shaft to minimize the impact in case the elevator cable breaks. A loaded car has mass 480kg, and its maximum height above the spring is 11.8m. In order to minimize the shock, the maximum acceleration of the car after hitting the spring is 4g. Find the spring constant, k.

Question

A large spring is placed at the bottom of an elevator shaft to minimize the impact in case the elevator cable breaks. A loaded car has mass 480kg, and its maximum height above the spring is 11.8m. In order to minimize the shock, the maximum acceleration of the car after hitting the spring is 4g. Find the spring constant, k.

Answer 1

F = kx

ma = kx
480kg * (4*9.81) = k * 11.8m
k = 1596.20339

Answer 2

2 equations:

Work spring = work of gravity: 0.5kx^2 = mgh
Force spring = Force of gravity + Force of car: kx = mg + m(4g)

Use the force equation to find that kx = 5mg, then substitute that into the work equation to get 0.5(5mg)x = mgh. Do some math and get x = h / 1.5. Substitute this answer back into the force equation: k = 5mg / (11.8 / 1.5) = 23520 / 7.867; k = 2989.7 N/m.

Answer 3

EDIT: h = 11.8 + x, x being the displacement of the spring once the car lands on it. Otherwise you'll get 2.5 instead of 1.5 when solving for x.

Answer 4

To find the spring constant, k, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

First, let's find the maximum potential energy the car has at the top before hitting the spring. We can use the equation for gravitational potential energy:

Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)

PE = 480 kg * 9.8 m/s^2 * 11.8 m
PE = 56410.4 J

Next, let's find the maximum force exerted on the spring. This force is equal to the weight of the car plus the force needed to resist the acceleration.

Force (F) = mass (m) * acceleration (a)

We are given that the maximum acceleration is 4g. Therefore, the magnitude of the acceleration is 4 times the acceleration due to gravity:

a = 4 * g
a = 4 * 9.8 m/s^2
a = 39.2 m/s^2

F = 480 kg * 39.2 m/s^2
F = 18816 N

Since the force exerted by the spring is proportional to its displacement, we have:

F = k * x

where k is the spring constant, and x is the displacement of the spring. In this case, the displacement is equal to the maximum height above the spring, which is 11.8 m.

18816 N = k * 11.8 m

Finally, we can solve for the spring constant, k:

k = 18816 N / 11.8 m
k ≈ 1596.61017 N/m

Therefore, the spring constant, k, is approximately 1596.61017 N/m.

F = kx

2990 N/m

2 equations:

EDIT: h = 11.8 + x, x being the displacement of the spring once the car lands on it. Otherwise you'll get 2.5 instead of 1.5 when solving for x.

To find the spring constant, k, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.