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Period of a Mass on a Spring

The period of a mass m on a spring of spring constant k can be calculated as [latex]\text{T}=2\pi \sqrt{\frac{\text{m}}{\text{k}}}[/latex].

Learning Objectives

Identify parameters necessary to calculate the period and frequency of an oscillating mass on the end of an ideal spring

Key Takeaways

Key Points

If an object is vibrating to the right and left, then it must have a leftward force on it when it is on the right side, and a rightward force when it is on the left side.

The restoring force causes an oscillating object to move back toward its stable equilibrium position, where the net force on it is zero.

The simplest oscillations occur when the restoring force is directly proportional to displacement. In this case the force can be calculated as F=-kx, where F is the restoring force, k is the force constant, and x is the displacement.

The motion of a mass on a spring can be described as Simple Harmonic Motion (SHM): oscillatory motion that follows Hooke's Law.

The period of a mass on a spring is given by the equation [latex]\text{T}=2\pi \sqrt{\frac{\text{m}}{\text{k}}}[/latex]

Key Terms

Restoring force: A variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system

amplitude: The maximum absolute value of some quantity that varies.

Understanding the Restoring Force

Newton's first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. It is important to understand how the force on the object depends on the object's position. If an object is vibrating to the right and left, then it must have a leftward force on it when it is on the right side, and a rightward force when it is on the left side. In one dimension, we can represent the direction of the force using a positive or negative sign, and since the force changes from positive to negative there must be a point in the middle where the force is zero. This is the equilibrium point, where the object would stay at rest if it was released at rest. It is common convention to define the origin of our coordinate system so that x equals zero at equilibrium.

Consider, for example, plucking a plastic ruler shown in the first figure. The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces (e.g., friction) dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.

Hooke's Law

The simplest oscillations occur when the restoring force is directly proportional to displacement. The name that was given to this relationship between force and displacement is Hooke's law:

[latex]\text{F}=\text{kx}[/latex]

Here, F is the restoring force, x is the displacement from equilibrium or deformation, and k is a constant related to the difficulty in deforming the system (often called the spring constant or force constant). Remember that the minus sign indicates the restoring force is in the direction opposite to the displacement. The force constant k is related to the rigidity (or stiffness) of a system-the larger the force constant, the greater the restoring force, and the stiffer the system. The units of k are newtons per meter (N/m). For example, k is directly related to Young's modulus when we stretch a string. A typical physics laboratory exercise is to measure restoring forces created by springs, determine if they follow Hooke's law, and calculate their force constants if they do.

Mass on a Spring

A common example of an objecting oscillating back and forth according to a restoring force directly proportional to the displacement from equilibrium (i.e., following Hooke's Law) is the case of a mass on the end of an ideal spring, where "ideal" means that no messy real-world variables interfere with the imagined outcome.

The motion of a mass on a spring can be described as Simple Harmonic Motion (SHM), the name given to oscillatory motion for a system where the net force can be described by Hooke's law. We can now determine how to calculate the period and frequency of an oscillating mass on the end of an ideal spring. The period T can be calculated knowing only the mass, m, and the force constant, k:

[latex]\text{T}=2\pi \sqrt{\frac{\text{m}}{\text{k}}}[/latex]

When dealing with [latex]\text{f}=1/\text{T}[/latex], the frequency is given by:

[latex]\text{f}=\frac{1}{2\pi} \sqrt{\frac{\text{k}}{\text{m}}}[/latex]

We can understand the dependence of these equations on m and k intuitively. If one were to increase the mass on an oscillating spring system with a given k, the increased mass will provide more inertia, causing the acceleration due to the restoring force F to decrease (recall Newton's Second Law: [latex]\text{F}=\text{ma}[/latex]). This will lengthen the oscillation period and decrease the frequency. In contrast, increasing the force constant k will increase the restoring force according to Hooke's Law, in turn causing the acceleration at each displacement point to also increase. This reduces the period and increases the frequency. The maximum displacement from equilibrium is known as the amplitude X.

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The period of a mass m on a spring of spring constant k can be calculated as [latex]\text{T}=2\pi \sqrt{\frac{\text{m}}{\text{k}}}[/latex]. Identify parameters necessary to calculate the period and frequency of an oscillating mass on the end of an ideal spring Key Points If an object is vibrating to the right and left, then it must have a leftward force on it when it is on the right side, and a rightward force when it is on the left side.

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