Introduction

In this exercise, we are given that an impulsive force gives an initial velocity of -1.0 m/s to a mass in the unstretched spring position. We need to determine the amplitude of motion for this system. Let's break down the process step by step:

Step 1: Understanding the Problem

Before we start solving the problem, let's understand the concepts involved. This problem deals with a mass-spring system, where a mass is attached to a spring and subjected to an impulsive force. The system will oscillate back and forth around the equilibrium position.

Step 2: Recall the Equation of Motion

The equation of motion for a mass-spring system is given by:

m * d²x/dt² = -k * x

Where:
m = mass of the object
x = displacement of the object from equilibrium position
k = spring constant

Step 3: Analyzing the Initial Conditions

We are given that the initial velocity of the mass is -1.0 m/s. This means that at t=0, the mass is moving in the negative direction. In terms of the equation of motion, this corresponds to an initial condition of:

dx/dt = -1.0 m/s

Step 4: Applying Initial Conditions

To solve the equation of motion, we need to apply the initial conditions. We integrate the equation of motion with respect to time, and then substitute the initial values to find the amplitude.

Step 5: Solving the Equation of Motion

Integrating the equation of motion, we get:

m * ∫d²x/dt² dt = -k * ∫x dt

Integrating both sides, we have:

m * ∫d²x/dt² dt = -k * ∫x dt

This gives us:

m * (dx/dt) = -k * x

Since we know dx/dt = -1.0 m/s, we can substitute this value into the equation:

m * (-1.0 m/s) = -k * x

Simplifying, we get:

m = k * x

Step 6: Finding the Amplitude

From the equation m = k * x, we can see that the amplitude of motion is directly proportional to the mass and inversely proportional to the spring constant. Therefore, to find the amplitude, we need to know the values of mass and spring constant.

Conclusion

In conclusion, the amplitude of motion in a mass-spring system can be found by applying the initial conditions and solving the equation of motion. In this problem, we were given an initial velocity of -1.0 m/s. By substituting this value into the equation and knowing the mass and spring constant, we can determine the amplitude of motion.

M=2kg,K=1200N/m