A 11.6 kg weather rocket generates a thrust of 215.0 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 399.0 N/m, is anchored to the ground. Initially, before the engine is ignited, the rocket sits at rest on top of the spring. How much is the spring compressed?

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To find out how much the spring is compressed, we need to calculate the force exerted by the spring and compare it to the weight of the rocket.

The force exerted by the spring can be calculated using Hooke's Law:

F_spring = k * x

Where:
F_spring is the force exerted by the spring
k is the spring constant
x is the displacement of the spring (compression)

In this case, the force exerted by the spring is equal to the weight of the rocket, which is given by:

F_weight = m * g

Where:
m is the mass of the rocket
g is the acceleration due to gravity (approximately 9.8 m/s^2)

We can set these two forces equal to each other and solve for x:

k * x = m * g

Substituting the given values gives us:

399.0 N/m * x = 11.6 kg * 9.8 m/s^2

Simplifying the equation:

x = (11.6 kg * 9.8 m/s^2) / 399.0 N/m

x ≈ 0.285 m

Therefore, the spring is compressed by approximately 0.285 meters.

To determine how much the spring is compressed, we need to analyze the forces acting on the rocket and the spring.

Given:
Mass of the rocket (m) = 11.6 kg
Thrust generated by the rocket (F) = 215.0 N
Spring constant (k) = 399.0 N/m

We can start by finding the net force acting on the rocket. The net force can be calculated using Newton's second law:

Net force (F_net) = ma

Where,
m = mass of the rocket
a = acceleration of the rocket

Since the rocket is initially at rest, its acceleration (a) is zero. Therefore, the net force (F_net) is also zero:

F_net = 0

Next, we need to consider the forces acting on the rocket and spring system. The forces involved are the gravitational force (mg) acting downward, the thrust force (F) acting upward, and the spring force (Fs) acting downward when the spring is compressed.

The gravitational force can be calculated as:

mg = mass of the rocket * acceleration due to gravity

Using the value of acceleration due to gravity (g) = 9.8 m/s^2:

mg = 11.6 kg * 9.8 m/s^2

Next, let's analyze the forces when the spring is compressed. At equilibrium, the sum of the forces acting on the rocket is zero:

F + Fs - mg = 0

Simplifying the equation by substituting the values we know:

215.0 N + Fs - (11.6 kg * 9.8 m/s^2) = 0

We can solve this equation to find the spring force (Fs):

Fs = (11.6 kg * 9.8 m/s^2) - 215.0 N

Once we know the spring force (Fs), we can calculate the compression of the spring using Hooke's law:

Fs = -kx

Where,
Fs = spring force
k = spring constant
x = compression of the spring

Rearranging the equation, we can solve for x:

x = -Fs / k

Substituting the values we have:

x = -((11.6 kg * 9.8 m/s^2) - 215.0 N) / 399.0 N/m

Calculating the value:

x ≈ -2.07 m

Since we are interested in the magnitude of the compression, we take the absolute value:

x ≈ 2.07 m

Therefore, the spring is compressed by approximately 2.07 meters.