doPhysics · Oscillations
Dynamics of SHM
The forces behind simple harmonic motion, and the period each one gives. Springs horizontal, vertical and in combination, and two blocks on one spring. The simple pendulum, also in a lift, and pendulums that swing and twist. A liquid in a U-tube, a floating cylinder, a tunnel through the Earth, and the energy of SHM.
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Why this chapter
Why this chapter exists
A block on a spring, a child on a swing and a disc on a wire look like different problems. So do water in a U-tube and a ball dropped into a tunnel through the Earth. Each has its own force: a spring, gravity along an arc, a twist, the weight of a liquid, buoyancy, gravity inside the Earth. Yet for small displacements every one of these forces is a straight line through the origin, \(F = -kx\). Once you see that line, the period follows in two steps. This chapter finds the line in each body, and then looks at the same motion through its energy.
You will learn
By the end of this chapter you will
- Explain what a restoring force is, and derive \(T = 2\pi\sqrt{m/k}\) from \(F = -kx\) and Newton's second law.
- Solve any oscillator in two steps: find \(k_{\text{eff}}\) for a small displacement, then read \(\omega^2 = k_{\text{eff}}/m_{\text{eff}}\).
- Show that a constant force shifts the mean position of a spring but not its period, and use \(T = 2\pi\sqrt{e/g}\) for a hanging mass.
- Find the period for springs in series, in parallel and cut, and for two blocks on one spring with the reduced mass.
- Derive \(T = 2\pi\sqrt{l/g}\) for a simple pendulum, and use \(g_{\text{eff}}\) in a lift, an accelerating car and free fall.
- Find the period of a physical pendulum and of a torsional pendulum from a restoring torque.
- Find the period of a liquid in a U-tube, a floating cylinder and a ball in a tunnel through the Earth.
- Follow the kinetic, potential and total energy of SHM with position and time, and find \(k\) from \(U''(x_0)\) by the energy method.
Before you begin
Before you begin
From Kinematics of SHM you should know \(a = -\omega^2x\), the amplitude, the period \(T = 2\pi/\omega\) and \(x = A\sin(\omega t + \phi)\). From Forces and Newton's Laws and Objects in Equilibrium you should know \(F = ma\), free-body diagrams and Hooke's law. From Potential Energy and Conservation of Energy you should know \(\tfrac12 kx^2\), springs in series and in parallel, and stable equilibrium. The chapter Rigid Bodies and Moment of Inertia gives the moment of inertia and the parallel axis theorem used in lesson 5.
Lessons
1
The Restoring Force and the Period
What makes a body return. A force that points to the mean position and grows with the displacement, \(F = -kx\), turns Newton's second law into the law of SHM. The period follows as \(T = 2\pi\sqrt{m/k}\), and it does not depend on the amplitude.
A restoring force points to the meanNewtons second law gives the periodThe period does not depend on the amplitudeTwo steps for every oscillator
2
The Spring: Horizontal and Vertical
The block on a spring is the model oscillator. On a smooth floor its restoring force is the spring force itself. A steady extra force, such as the weight of a hanging block, moves the mean position but not the period. So a hanging mass has \(T = 2\pi\sqrt{e/g}\).
The block on a horizontal springA constant force shifts the meanThe vertical spring
3
Springs in Combination
Two springs on one block act as one spring of a new constant. In parallel the constants add, in series their reciprocals add, and a cut piece is stiffer. Two blocks on one spring oscillate with the reduced mass, because one point of the spring never moves.
Springs in parallel addSprings in seriesA cut spring is stifferTwo blocks and the reduced mass
4
The Simple Pendulum
Gravity along an arc. The component \(mg\sin\theta\) of the weight pulls the bob back to the lowest point. For small angles it becomes a linear restoring force. So \(T = 2\pi\sqrt{l/g}\), with no mass in it. In a lift or an accelerating car, \(g\) is replaced by the effective gravity.
Gravity along the arcSmall angles give shmThe seconds pendulumPendulum in an accelerating frame
5
Pendulums That Twist and Swing
When a rigid body turns, the restoring force becomes a torque and the mass becomes a moment of inertia. The two steps stay the same. A physical pendulum swinging about a pivot has \(T = 2\pi\sqrt{I/mgd}\), and a disc twisting on a wire has \(T = 2\pi\sqrt{I/C}\).
A restoring torqueThe physical pendulumThe torsional pendulum
6
Liquids and the Earth as Springs
Three oscillators with no spring at all. A liquid column in a U-tube, a floating cylinder and a ball inside the Earth. Each is a linear restoring force, so each gives SHM in the same two steps.
Liquid in a u tubeA floating cylinderA tunnel through the earth
7
Energy in SHM
An oscillator trades kinetic energy for potential energy and back, twice every period, while their sum \(\tfrac12 kA^2\) stays fixed. The two are equal at \(x = \pm A/\sqrt{2}\). Turned round, the energy gives a second way to find \(\omega\): near any stable minimum of \(U(x)\), \(k\) is the curvature \(U''(x_0)\).
Kinetic and potential energy in shmThe total energy is constantWhere kinetic equals potentialThe energy method
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- buoyant force
- The upward force of a liquid on a body in it, equal to the weight of the liquid the body pushes aside.
- centre of oscillation
- The point on the line from the pivot through the centre of mass of a physical pendulum, at the equivalent length from the pivot.
- centre of percussion
- The point of a pivoted body where a blow produces no jolt at the pivot. It coincides with the centre of oscillation.
- effective force constant
- The constant \(k_{\text{eff}}\) in the restoring force \(F = -k_{\text{eff}}x\) of any oscillator, whatever produces the force.
- effective gravity
- The gravity felt in an accelerating frame, \(g_{\text{eff}} = g - a\) as vectors. A pendulum in the frame swings with \(T = 2\pi\sqrt{l/g_{\text{eff}}}\).
- energy method
- Finding the angular frequency of small oscillations from the potential energy: near a stable minimum \(x_0\), \(k_{\text{eff}} = U''(x_0)\) and \(\omega^2 = U''(x_0)/m\).
- equivalent length
- The length of the simple pendulum that has the same period as a given physical pendulum, \(L_{\text{eq}} = d + k^2/d\).
- force constant
- The restoring force per unit displacement, \(k\) in \(F = -kx\), in N/m. For a spring it is the spring constant; for other oscillators it is the effective force constant.
- isochronous
- Having a period that does not depend on the amplitude. Every oscillator with a linear restoring force \(F = -kx\) is isochronous.
- kinetic energy
- The energy a body has because it moves, \(K = \tfrac12 mv^2\).
- mean position
- The equilibrium position about which a body oscillates. The net force on the body is zero there.
- mechanical energy
- The sum of the kinetic and the potential energy of a system. For SHM without friction it is \(\tfrac12 kA^2\) and stays constant.
- physical pendulum
- A rigid body swinging under its own weight about a fixed horizontal axis away from its centre of mass. It is also called a compound pendulum, and \(T = 2\pi\sqrt{I/mgd}\).
- radius of gyration
- The distance \(k\) at which the whole mass of a body would have to sit to give the same moment of inertia: \(I = mk^2\).
- reduced mass
- The mass \(\mu = m_1m_2/(m_1 + m_2)\) that takes the place of \(m\) when two bodies oscillate on one spring with neither end fixed. It is smaller than either mass.
- restoring force
- A force that always points back towards the mean position, from whichever side the body is displaced.
- restoring torque
- A torque that always turns a body back towards its equilibrium angle. When it is proportional to the angle, \(\tau = -k_\theta\theta\), the body turns in SHM.
- seconds pendulum
- A simple pendulum whose period is exactly 2 s, so that each swing from side to side takes one second. Its length is \(g/\pi^2\), about 0.99 m.
- simple pendulum
- A small heavy bob on a light inextensible string, swinging about a fixed point. For small swings \(T = 2\pi\sqrt{l/g}\).
- spring-block system
- A block fixed to one end of a light spring whose other end is held still. On a smooth floor it oscillates with \(T = 2\pi\sqrt{m/k}\).
- springs in parallel
- Springs that share one displacement of the body they hold. Their forces add, so the pair acts as one spring of constant \(k_1 + k_2\).
- springs in series
- Springs joined end to end, so that one force acts through each. Their stretches add, so \(1/k_s = 1/k_1 + 1/k_2\).
- static extension
- The stretch \(e\) of a spring when a hanging load rests on it, given by \(ke = mg\). The load oscillates about this stretched position with \(T = 2\pi\sqrt{e/g}\).
- torsion constant
- The restoring torque of a twisted wire per radian of twist, \(C\) in \(\tau = -C\theta\), in N m/rad.
- torsional pendulum
- A body hung on a wire that turns to and fro about the wire, twisting it. Its period is \(T = 2\pi\sqrt{I/C}\).
- U-tube
- A bent tube of uniform bore with two upright arms, open at both ends. A liquid column of length \(L\) in it oscillates with \(T = 2\pi\sqrt{L/2g}\).