Energy of mass on spring. A mass of 50 g on the end of a certain spring executes simple harmoni

step 1 Hello, the question is taken from physics and the question is energy of a mass on spring. So a m

Energy of mass on spring. A mass of 50 g on the end of a...

Energy of mass on spring. A mass of $50 \mathrm{~g}$ on the end of a certain spring executes simple harmonic motion according to the equation $x=2 \sin 10 t$ where $x$ is in centimeters and $t$ is in seconds. (a) Find the spring constant $C$. (b) Find the maximum kinetic energy. (c) What are the maximum potential energy and the total energy?

Energy of mass on spring. A mass of $50 \mathrm{~g}$ on the end of a certain spring executes simple harmonic motion according to the equation $x=2 \sin 10 t$ where $x$ is in centimeters and $t$ is in seconds.
(a) Find the spring constant $C$.
(b) Find the maximum kinetic energy.
(c) What are the maximum potential energy and the total energy?

Key Concepts

Kinetic Energy

Kinetic energy in a mass-spring system is the energy associated with the motion of the mass and varies with its speed. During SHM, the kinetic energy reaches its maximum when the mass passes through the equilibrium position, where the speed is highest. It is calculated using the basic kinetic energy formula involving mass and velocity.

Conservation of Energy

The principle of conservation of energy states that in a closed and frictionless system, the total energy remains constant over time. In the context of a mass-spring oscillator, the total mechanical energy is the sum of kinetic and potential energies, with energy continuously exchanged between these two forms as the object oscillates, while the overall total energy stays fixed.

Spring Constant

The spring constant, often denoted by k or C, quantifies the stiffness of the spring. According to Hooke’s Law, the restoring force exerted by the spring is proportional to the displacement, and this proportionality constant is the spring constant. In a mass-spring system undergoing SHM, it relates to the angular frequency by the formula ?² = k/m, linking the system’s physical properties to its motion.

Potential Energy

Potential energy in a spring system is stored due to the displacement of the mass from the equilibrium position. It is given by the formula (1/2)kx², where k is the spring constant and x is the displacement. In SHM, the potential energy is at its maximum when the displacement is at its peak (the amplitude) and is zero when the mass passes through equilibrium.

Angular Frequency

Angular frequency, typically denoted by ?, represents how fast the oscillation occurs in radians per unit time. In the context of SHM, it determines the rate at which the sinusoidal function oscillates and is directly linked to the physical characteristics of the system, such as the mass and spring constant when dealing with a mass-spring system.

Amplitude

Amplitude in the context of SHM is the maximum displacement from the equilibrium position reached by the oscillating object. It provides a measure of the energy in the system since the potential energy stored in the spring at maximum displacement is proportional to the square of the amplitude.

Simple Harmonic Motion

Simple harmonic motion (SHM) is the type of oscillatory motion experienced by a system where the restoring force is directly proportional to the displacement from the equilibrium position. This concept is fundamental in understanding systems like a mass attached to a spring, and it is characterized by sinusoidal functions that describe the position, velocity, and acceleration of the oscillating body over time.

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