Physics Review: Oscillations, Work, Energy, and Power

Comprehensive practice flashcards covering Oscillations (SHM, period, frequency, force law, energies, pendulum) and Work, Energy, and Power (scalar product, work-energy theorem, variable forces, conservative systems, vertical circular motion, springs, power, and collisions).

What is the key distinction between periodic motion and oscillatory motion?

Periodic motion repeats itself at regular intervals of time. Oscillatory motion is a specific periodic motion in which a body moves to and fro about a mean (equilibrium) position. Every oscillatory motion is periodic, but every periodic motion (such as uniform circular planetary motion) is not oscillatory.

How are period ($T$) and frequency ($\nu$) defined, and what is the relation between them?

The period ($T$) is the smallest interval of time after which motion repeats (SI unit: $\text{s}$). The frequency ($\nu$) is the number of complete oscillations per unit time, given by $\nu = \frac{1}{T}$. The SI unit of frequency is the hertz ($\text{Hz}$), where $1\,\text{Hz} = 1\,\text{s}^{-1}$.

On an average, a human heart beats $75$ times in a minute. What are its frequency and period?

The beat frequency is $\nu = \frac{75}{60\,\text{s}} = 1.25\,\text{Hz}$. The time period is $T = \frac{1}{\nu} = \frac{1}{1.25\,\text{s}^{-1}} = 0.8\,\text{s}$.

Why is the term 'displacement' generalized beyond spatial position in the context of oscillations?

Displacement refers broadly to the change with time of any physical property under consideration. Examples include the angular position ($\theta$) of a pendulum, alternating voltage across a capacitor, pressure variations in sound waves, and electric/magnetic fields in light waves.

What fundamental theorem proved by Jean Baptiste Joseph Fourier relates to periodic functions?

Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients.

How can the linear combination $f(t) = A\sin(\omega t) + B\cos(\omega t)$ be written as a single sinusoidal function?

It can be written as $f(t) = D\sin(\omega t + \phi)$, where $D = \sqrt{A^2 + B^2}$ and $\phi = \tan^{-1}\left(\frac{B}{A}\right)$.

Why is the function $f(t) = \sin(\omega t) + \cos(2\omega t) + \sin(4\omega t)$ periodic, and what is its fundamental period?

Each term is periodic with individual periods of $T_0 = \frac{2\pi}{\omega}$, $\frac{T_0}{2}$, and $\frac{T_0}{4}$. Since $T_0$ is an integral multiple of the other periods, the smallest interval after which the entire sum repeats is $T = \frac{2\pi}{\omega}$.

What is the defining equation of displacement in simple harmonic motion (SHM)?

$x(t) = A\cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency ($\omega = \frac{2\pi}{T}$), $(\omega t + \phi)$ is the time-dependent phase, and $\phi$ is the phase constant (initial phase at $t = 0$).

What is the expression for the instantaneous velocity $v(t)$ in SHM, and what is its phase relative to displacement?

$v(t) = \frac{dx}{dt} = -\omega A\sin(\omega t + \phi)$. The velocity amplitude is $v_m = \omega A$, and the velocity plot has a phase difference of $\frac{\pi}{2}$ ahead of the displacement plot.

What is the expression for the instantaneous acceleration $a(t)$ in SHM, and how is it directed?

$a(t) = \frac{dv}{dt} = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x(t)$. The acceleration is always directly proportional to displacement and directed towards the mean position (opposite to displacement), having a phase difference of $\pi$ relative to displacement.

What is the force law for a linear harmonic oscillator?

$F(t) = -kx(t)$, where $k = m\omega^2$ is the spring constant or force constant. The angular frequency is $\omega = \sqrt{\frac{k}{m}}$, and the period is $T = 2\pi\sqrt{\frac{m}{k}}$.

For the system shown with a mass $m$ connected between two identical fixed springs of constant $k$, what is the net restoring force and the time period?

The net restoring force is $F = -2kx$, and the period of oscillation is $T = 2\pi\sqrt{\frac{m}{2k}}$.

What are the expressions for the kinetic energy ($K$) and potential energy ($U$) in SHM, and what are their repetition periods?

$K(t) = \frac{1}{2}kA^2\sin^2(\omega t + \phi)$ and $U(t) = \frac{1}{2}kx^2 = \frac{1}{2}kA^2\cos^2(\omega t + \phi)$. Both energies oscillate between zero and $\frac{1}{2}kA^2$ with a period of $\frac{T}{2}$.

What is the total mechanical energy $E$ of a linear simple harmonic oscillator?

$E = K + U = \frac{1}{2}kA^2$. It is constant, independent of time $t$ and displacement $x$.

In the free-body diagram of the simple pendulum shown, which component of the gravitational force provides the restoring torque about the support?

The tangential component of gravity, $mg\sin(\theta)$, provides the restoring torque $\tau = -L(mg\sin(\theta))$. The radial component $mg\cos(\theta)$ and string tension $T$ produce zero torque about the support.

Under what condition does a simple pendulum execute simple harmonic motion, and what is its period formula?

It executes SHM for small angular displacements where $\sin(\theta) \approx \theta$ (in radians). Its time period is given by $T = 2\pi\sqrt{\frac{L}{g}}$, which is independent of the bob's mass.

What is a 'seconds pendulum', and what is its approximate length where $g = 9.8\,\text{m}\,\text{s}^{-2}$?

A seconds pendulum has a time period of $T = 2\,\text{s}$ (one second for each swing). Its length is $L = \frac{gT^2}{4\pi^2} = \frac{9.8 \times 4}{4\pi^2} \approx 1\,\text{m}$.

How is the scalar (dot) product of vectors $\mathbf{A}$ and $\mathbf{B}$ defined, and how is it expressed in Cartesian components?

$\mathbf{A}\cdot\mathbf{B} = AB\cos(\theta)$. In component form, $\mathbf{A}\cdot\mathbf{B} = A_x B_x + A_y B_y + A_z B_z$.

Under what three conditions is the work done on an object by a force equal to zero?

(1) The displacement is zero ($d = 0$), (2) the applied force is zero ($F = 0$), or (3) the force and displacement are mutually perpendicular ($\theta = 90^\circ$, so $\cos(90^\circ) = 0$).

A cyclist comes to a skidding stop over $10\,\text{m}$ due to a road friction force of $200\,\text{N}$. What is the work done on the cycle by the road, and on the road by the cycle?

Work done on the cycle by the road is $W = Fd\cos(180^\circ) = 200 \times 10 \times (-1) = -2000\,\text{J}$. The work done on the road by the cycle is $0\,\text{J}$ because the road undergoes no displacement.

State the Work-Energy Theorem for a particle.

The change in kinetic energy of a particle is equal to the work done on it by the net force: $K_f - K_i = W_{\text{net}}$.

A raindrop of mass $1.00\,\text{g}$ falls from $1.00\,\text{km}$ and hits the ground at $50.0\,\text{m}\,\text{s}^{-1}$ ($g = 10\,\text{m/s}^2$). What is the work done by gravity and by the resistive force?

Work done by gravity is $W_g = mgh = 10^{-3}\,\text{kg} \times 10\,\text{m/s}^2 \times 10^3\,\text{m} = 10.0\,\text{J}$. Change in kinetic energy is $\Delta K = \frac{1}{2}(10^{-3}\text{ kg})(50.0\text{ m/s})^2 = 1.25\,\text{J}$. By the work-energy theorem, the resistive force does $W_r = \Delta K - W_g = 1.25 - 10.0 = -8.75\,\text{J}$.

How is work done calculated for a variable force $F(x)$ acting along the $x$-axis from position $x_i$ to $x_f$?

$W = \int_{x_i}^{x_f} F(x)\,dx$, which corresponds to the definite area under the $F(x)$ versus $x$ curve between $x_i$ and $x_f$.

What are three defining properties of a conservative force?

(1) The force can be derived from a scalar potential energy function: $F(x) = -\frac{dV}{dx}$. (2) The work done depends only on the initial and final endpoints, not on the path. (3) The work done around any closed path is zero ($\oint \mathbf{F}\cdot d\mathbf{r} = 0$).

What is the principle of conservation of mechanical energy?

The total mechanical energy of a system ($E = K + V$) remains constant throughout the motion if the forces doing work on it are conservative: $K_i + V(x_i) = K_f + V(x_f)$.

In the potential and kinetic energy plots for a block on a spring shown, how do $V$, $K$, and total energy $E$ vary with displacement?

Potential energy $V(x) = \frac{1}{2}kx^2$ is parabolic and minimum ($0$) at $x = 0$, reaching maximum at $x = \pm x_m$. Kinetic energy $K(x)$ is complementary, peaking at $x = 0$ and dropping to zero at $x = \pm x_m$. Their sum $E = K + V$ is represented by a constant horizontal line.

For a bob suspended on a string of length $L$ to complete a vertical circle without slacking before the topmost point $C$, what are the minimum speeds at lowest point $A$ and highest point $C$?

The minimum speed at the lowest point $A$ is $v_0 = \sqrt{5gL}$, and the speed at the topmost point $C$ is $v_C = \sqrt{gL}$.

What is the ratio of kinetic energies ($K_B/K_C$) at the horizontal point $B$ and topmost point $C$ for a bob completing a vertical circle on a string?

Since $v_B = \sqrt{3gL}$ and $v_C = \sqrt{gL}$, the ratio of kinetic energies is $\frac{K_B}{K_C} = \frac{\frac{1}{2}m(3gL)}{\frac{1}{2}m(gL)} = \frac{3}{1}$.

How much work is done by the spring force when an ideal spring of constant $k$ is extended from $x_i$ to $x_f$?

$W_s = -\int_{x_i}^{x_f} kx\,dx = \frac{1}{2}kx_i^2 - \frac{1}{2}kx_f^2$.

How is instantaneous power defined in terms of work, and how is it related to force and velocity?

Instantaneous power is $P = \frac{dW}{dt}$. In terms of force and velocity, $P = \mathbf{F}\cdot\mathbf{v} = Fv\cos(\theta)$. Its SI unit is the watt ($\text{W}$), where $1\,\text{W} = 1\,\text{J}\,\text{s}^{-1}$.

How many watts are in one horsepower ($1\,\text{hp}$), and how many joules are in one kilowatt-hour ($1\,\text{kWh}$)?

$1\,\text{hp} = 746\,\text{W}$, and $1\,\text{kWh} = 1000\,\text{W} \times 3600\,\text{s} = 3.6 \times 10^6\,\text{J}$.

What distinguishes elastic, inelastic, and completely inelastic collisions?

Linear momentum is conserved in all collisions. In an elastic collision, total kinetic energy is conserved before and after impact. In an inelastic collision, some kinetic energy is lost to heat, sound, or deformation. In a completely inelastic collision, the colliding bodies stick together and move with a common velocity.

In a 1D elastic collision where mass $m_1$ with velocity $v_{1i}$ hits stationary mass $m_2$, what are the final velocities $v_{1f}$ and $v_{2f}$?

$v_{1f} = \frac{m_1 - m_2}{m_1 + m_2}v_{1i}$ and $v_{2f} = \frac{2m_1}{m_1 + m_2}v_{1i}$.

What happens to the velocities when two equal masses ($m_1 = m_2$) undergo a head-on elastic collision with the second mass initially at rest?

They completely exchange velocities: the incident particle comes to rest ($v_{1f} = 0$), and the target particle moves off with the incident particle's initial speed ($v_{2f} = v_{1i}$).

When two equal masses undergo a glancing elastic collision with one target initially at rest, what is the angle between their final paths?

After the collision, the two bodies move at right angles ($90^\circ$) to each other ($\theta_1 + \theta_2 = 90^\circ$).