Problem 25 - College Physics

Question

\(\bullet\) Write the general equation of motion for a mass that is on a horizontal frictionless surface and is connected to a spring at equilibrium (a) if the mass is initially pulled in the \(+x\) axis from the spring (stretched) and released, and (b) if the mass is pushed in the \(-x\) axis toward the spring (compressed) and released.

Step-by-Step Solution

Answer

(a) \(x(t) = A \cos(\omega t)\); (b) \(x(t) = -A \cos(\omega t)\).

Step 1: Define the System

Consider a mass-spring system where a mass, \(m\), is attached to a horizontal spring with spring constant \(k\). This system is frictionless.

Step 2: Principle of Simple Harmonic Motion

The motion of the mass on a spring is an example of simple harmonic motion (SHM). The equation of motion for SHM is given by \(F = -kx\), where \(x\) is the displacement from equilibrium.

Step 3: Derive the Equation of Motion

Using Newton's second law, \(F = ma\), we can set \(ma = -kx\). Rearranging gives the differential equation \(\frac{d^2x}{dt^2} = -\frac{k}{m}x\). This is the standard form for SHM.

Step 4: Construct the General Solution for SHM

The general solution of the differential equation for SHM is \(x(t) = A \cos(\omega t + \phi)\), where \(A\) is the amplitude, \(\omega = \sqrt{\frac{k}{m}}\) is the angular frequency, and \(\phi\) is the phase constant.

Step 5: Solve for Initial Conditions (a) Stretched Spring

For part (a), the spring is stretched, meaning the mass is pulled in the \(+x\) direction. Initially, \(x(0) = A\) and \(v(0) = 0\). Substituting these into the general solution, we find \(\phi = 0\), so the equation becomes \(x(t) = A \cos(\omega t)\).

Step 6: Solve for Initial Conditions (b) Compressed Spring

For part (b), the spring is compressed, meaning the mass is pushed towards the spring, \(-x\). Initially, \(x(0) = -A\) and \(v(0) = 0\). Substituting these into the general solution, we find \(\phi = \pi\), so the equation becomes \(x(t) = -A \cos(\omega t)\) or equivalently \(x(t) = A \cos(\omega t + \pi)\).

Key Concepts

Equation of Motion

In a typical mass-spring system, the equation of motion describes how the mass moves over time due to the spring force. This equation is fundamental to understanding simple harmonic motion (SHM). To begin, let's recall that the force exerted by a spring is given by Hooke’s Law:

• \( F = -kx \)

where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position. The negative sign indicates that the force exerted by the spring always acts in the opposite direction of the displacement. Applying Newton’s second law, which is \( F = ma \), we can set these equations equal to each other:

• \( ma = -kx \)

Here, \( a \) is the acceleration of the mass. Rearranging the equation gives us the familiar form of the differential equation for SHM:

• \( \frac{d^2x}{dt^2} = -\frac{k}{m}x \)

This differential equation forms the cog in the simple harmonic motion wheel. It tells us that the acceleration of the mass is directly proportional to the negative of its displacement, which ensures oscillatory motion.

Mass-Spring System

A mass-spring system consists of a mass attached to a spring, which can either stretch or compress. This system is a classic example of Simple Harmonic Motion (SHM). When the mass is on a horizontal frictionless surface, its motion is purely influenced by the spring's force. - If you pull the mass to stretch the spring (as in the case of the mass moving in the \(+x\) direction), the spring stores potential energy.- If you push the mass, compressing the spring (mass moving in the \(-x\) direction), the energy is again stored but starts as more compressed relative to the equilibrium point. When released from any initial position, the mass will oscillate about the equilibrium position due to the interplay of potential energy in the spring and the kinetic energy of the mass. Through these oscillations, the system follows the equation of motion derived from the spring force, leading to a predictable, cyclic pattern.

Differential Equation

The motion of a mass in a mass-spring system is described by a differential equation. This mathematical equation is crucial for explaining how displacement varies with time. The differential equation we derive from balancing forces in SHM is:

• \( \frac{d^2x}{dt^2} = -\frac{k}{m}x \)

This is a second-order differential equation, which means it includes the second derivative of \( x \) with respect to time, representing acceleration. Solving this differential equation gives us the general solution of Simple Harmonic Motion:

• \( x(t) = A \cos(\omega t + \phi) \)

Here, \( A \) represents amplitude, \( \omega = \sqrt{\frac{k}{m}} \) is the angular frequency derived from the spring constant and mass, and \( \phi \) is the phase constant. This solution describes how the displacement of the mass evolves over time under harmonic motion, ultimately embodying the essence of SHM in one clean equation.