Solving Mass on a Spring Homework: Step-by-Step Guide

[joeypeter]

Homework Statement

A 1.30 kg mass on a spring oscilates horizontally with little friction according to the following equation: x = 0.070cos(2.50t), where x is in meters and t in seconds.
1) Find the maximum energy stored in the spring during an oscillation.

2) Find the maximum velocity of the mass.

Homework Equations

I have no clue how to do this problem. Can someone show me step by step how to do it.

The Attempt at a Solution

 

[Snazzy]

The displacement function of simple harmonic motion is:

$$x(t) = Acos(\omega t +\phi )$$

There is an equation that relates the angular frequency to the mass and the spring constant, and the maximum energy of a SHM system is 1/2 kA^2.

The maximum velocity can be obtained by finding the derivative of the displacement function and determining what the maximum value of sin is.

 

[Moetasim]

I would like to provide a step-by-step guide on how to solve this problem. Firstly, we need to understand the given equation and its variables. The given equation represents the displacement of the mass on the spring, with x being the displacement in meters and t being the time in seconds.

Step 1: Understanding the Concept of Energy Stored in a Spring
To find the maximum energy stored in the spring, we need to understand the concept of potential energy in a spring. When a spring is stretched or compressed, it stores potential energy. The amount of potential energy stored in a spring is directly proportional to the displacement of the mass and the spring constant, which is a measure of the stiffness of the spring. Mathematically, the potential energy stored in a spring can be expressed as U = (1/2)kx^2, where k is the spring constant and x is the displacement.

Step 2: Finding the Spring Constant
To find the spring constant, we can use the given equation. As we know that the equation for simple harmonic motion is x = A cos(ωt), where A is the amplitude and ω is the angular frequency. Comparing this with the given equation, we can see that A = 0.070 m and ω = 2.50 rad/s. The spring constant can be calculated using the formula k = mω^2, where m is the mass of the object. Substituting the values, we get k = (1.30 kg)(2.50 rad/s)^2 = 8.125 N/m.

Step 3: Calculating the Maximum Energy
Now, we can use the formula U = (1/2)kx^2 to calculate the maximum energy stored in the spring. As we know that the displacement is given by x = 0.070cos(2.50t), we can calculate the maximum displacement by substituting t = 0. We get x = 0.070 m. Substituting this value in the formula, we get U = (1/2)(8.125 N/m)(0.070 m)^2 = 0.0201 J.

Step 4: Finding the Maximum Velocity
To find the maximum velocity of the mass, we can use the formula v = ωA, where ω is the angular frequency and A is the amplitude. Substituting the values, we get v