# Energy in a spring-mass system

In summary, the conversation discusses a horizontal spring-mass system with low friction, a spring stiffness of 220 N/m, and a mass of 0.5 kg. The system is released with an initial compression of 10 cm and an initial speed of 3 m/s. The maximum speed during the motion is unknown. The conversation then introduces energy dissipation of 0.02 J per cycle and asks for the average power input required to maintain a steady oscillation. The equations and attempts at solutions for determining the maximum stretch, angular frequency, and frequency are also mentioned.

## Homework Statement

A horizontal spring-mass system has low friction, spring stiffness 220 N/m, and mass 0.5 kg. The system is released with an initial compression of the spring of 10 cm and an initial speed of the mass of 3 m/s.

(b) What is the maximum speed during the motion?
(c) Now suppose that there is energy dissipation of 0.02 J per cycle of the spring-mass system. What is the average power input in watts required to maintain a steady oscillation?

## The Attempt at a Solution

(a) What is the maximum stretch during the motion?
I got .1745

## The Attempt at a Solution

(a) What is the maximum stretch during the motion?
I got .1745

I did not check this, but I will assume it is correct.

What is the angular frequency of the oscillation and hence what is the frequency?

## What is a spring-mass system?

A spring-mass system is a physical system that consists of a mass attached to a spring, which is fixed at one end and free to move at the other end. The mass and spring are connected in such a way that when the mass is displaced from its equilibrium position, the spring exerts a restoring force in the opposite direction, causing the mass to oscillate back and forth.

## What is the relationship between energy and a spring-mass system?

In a spring-mass system, energy is constantly being exchanged between potential energy stored in the spring and kinetic energy of the mass as it oscillates. As the mass moves away from the equilibrium position, it gains potential energy due to the stretched or compressed spring. As it moves back towards the equilibrium position, this potential energy is converted into kinetic energy. This constant exchange of energy allows the system to continue oscillating.

## What are the different types of energy in a spring-mass system?

The two main types of energy in a spring-mass system are potential energy and kinetic energy. Potential energy is stored in the spring when it is stretched or compressed, while kinetic energy is the energy of motion of the mass as it oscillates. There may also be other forms of energy present, such as thermal energy due to friction, but these are typically negligible in ideal systems.

## How does the mass and spring constant affect the energy in a spring-mass system?

The mass and spring constant both have a direct impact on the energy in a spring-mass system. A larger mass will result in a slower oscillation and therefore less energy being exchanged between potential and kinetic energy. A larger spring constant will result in a faster oscillation and more energy being exchanged. In general, a system with a higher spring constant will have more energy than one with a lower spring constant.

## What factors can affect the energy conservation in a spring-mass system?

In an ideal system, energy is conserved in a spring-mass system. However, factors such as friction, air resistance, and non-ideal behavior of the spring can cause some energy to be lost. Additionally, external forces acting on the mass, such as gravity, can also affect the energy conservation. These factors can cause the amplitude of the oscillation to decrease over time, resulting in a decrease in the total energy of the system.