University Physics Volume 1

# Summary

### 15.1Simple Harmonic Motion

• Periodic motion is a repeating oscillation. The time for one oscillation is the period T and the number of oscillations per unit time is the frequency f. These quantities are related by $f=1Tf=1T$.
• Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
• Maximum displacement is the amplitude A. The angular frequency $ωω$, period T, and frequency f of a simple harmonic oscillator are given by $ω=kmω=km$, $T=2πmk,andf=12πkmT=2πmk,andf=12πkm$, where m is the mass of the system and k is the force constant.
• Displacement as a function of time in SHM is given by$x(t)=Acos(2πTt+ϕ)=Acos(ωt+ϕ)x(t)=Acos(2πTt+ϕ)=Acos(ωt+ϕ)$.
• The velocity is given by $v(t)=−Aωsin(ωt+ϕ)=−vmaxsin(ωt+ϕ),wherevmax=Aω=Akmv(t)=−Aωsin(ωt+ϕ)=−vmaxsin(ωt+ϕ),wherevmax=Aω=Akm$.
• The acceleration is $a(t)=−Aω2cos(ωt+ϕ)=−amaxcos(ωt+ϕ)a(t)=−Aω2cos(ωt+ϕ)=−amaxcos(ωt+ϕ)$, where $amax=Aω2=Akmamax=Aω2=Akm$.

### 15.2Energy in Simple Harmonic Motion

• The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
• Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given by$U=12kx2.U=12kx2.$
• Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
$ETotal=12mv2+12kx2=12kA2=constant.ETotal=12mv2+12kx2=12kA2=constant.$
• The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
$|v|=km(A2−x2).|v|=km(A2−x2).$

### 15.3Comparing Simple Harmonic Motion and Circular Motion

• A projection of uniform circular motion undergoes simple harmonic oscillation.
• Consider a circle with a radius A, moving at a constant angular speed $ωω$. A point on the edge of the circle moves at a constant tangential speed of $vmax=Aωvmax=Aω$. The projection of the radius onto the x-axis is $x(t)=Acos(ωt+ϕ)x(t)=Acos(ωt+ϕ)$, where $(ϕ)(ϕ)$ is the phase shift. The x-component of the tangential velocity is $v(t)=−Aωsin(ωt+ϕ)v(t)=−Aωsin(ωt+ϕ)$.

### 15.4Pendulums

• A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about $15°15°$. The period of a simple pendulum is $T=2πLgT=2πLg$, where L is the length of the string and g is the acceleration due to gravity.
• The period of a physical pendulum $T=2πImgLT=2πImgL$ can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L.
• The period of a torsional pendulum $T=2πIκT=2πIκ$ can be found if the moment of inertia and torsion constant are known.

### 15.5Damped Oscillations

• Damped harmonic oscillators have non-conservative forces that dissipate their energy.
• Critical damping returns the system to equilibrium as fast as possible without overshooting.
• An underdamped system will oscillate through the equilibrium position.
• An overdamped system moves more slowly toward equilibrium than one that is critically damped.

### 15.6Forced Oscillations

• A system’s natural frequency is the frequency at which the system oscillates if not affected by driving or damping forces.
• A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
• The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.