| Step | Derivation/Formula | Reasoning |
|---|---|---|
| (a) | \[\alpha = \frac{\Delta\omega}{\Delta t}=\frac{-30-60}{6-2}=\frac{-90}{4}=-22.5\,\text{rad\,/s}^2\] | The interval from \(t=2\,\text{s}\) to \(t=6\,\text{s}\) is a straight line, so the slope (constant) gives the instantaneous angular acceleration at any time in that portion, including \(t=4\,\text{s}.\) |
| (b) | \[v=r\,\omega = 0.25\,\text{m}\times 60\,\text{rad\,/s}=15\,\text{m\,/s}\] | At \(t=1\,\text{s}\) the graph is flat at \(\omega = 60\,\text{rad\,/s}.\) Linear (rim) speed is \(v=r\omega.\) |
| (c) | \[\Delta\theta_{0\!\to 2}=\omega\,\Delta t = 60\,\text{rad\,/s}\times 2\,\text{s}=120\,\text{rad}\] | From 0–2 s the angular velocity is constant, so the area under the \(\omega\)-vs-\(t\) graph (a rectangle) is \(\omega\Delta t.\) |
| (d) | \[\omega_f = \omega_i + \alpha\,\Delta t = 60 + (-22.5)(2)=15\,\text{rad\,/s}\] \[\Delta\theta_{2\!\to 4}=\tfrac12(\omega_i+\omega_f)\,\Delta t = \tfrac12(60+15)\times 2 = 75\,\text{rad}\] |
The segment 2–4 s lies on the linear portion with constant \(\alpha.\) Use kinematics (or trapezoid area) with \(\omega_i=60\,\text{rad\,/s}\) and \(\omega_f=15\,\text{rad\,/s}.\) |
| (e) | \[\begin{aligned} \Delta\theta_{0\!\to 2}&=120\\[4pt] \Delta\theta_{2\!\to 6}&=\tfrac12(60+(-30))(4)=60\\[4pt] \Delta\theta_{6\!\to 8}&=(-30)(2)=-60\\[4pt] \Delta\theta_{8\!\to 10}&=\tfrac12(-30+0)(2)=-30\\[4pt] \theta_{\text{total}}&=120+60-60-30=90\,\text{rad} \end{aligned}\] |
Sum the signed areas (trapezoids/rectangles) for each time interval. Positive areas correspond to counter-clockwise rotation; negative areas to clockwise. |
| (f) | \[\Delta x = r\,\theta_{\text{total}} = 0.25\,\text{m}\times 90\,\text{rad}=22.5\,\text{m}\] | For rolling without slipping, the center of mass translates a linear distance equal to \(r\,\Delta\theta.\) |
| (g) | \[a_{\text{tan}} = r\,|\alpha| = 0.25\,\text{m}\times 22.5\,\text{rad\,/s}^2 = 5.6\,\text{m\,/s}^2\] | Magnitude of tangential acceleration is the product of radius and magnitude of angular acceleration at \(t=4\,\text{s}.\) |
| (h) | \[a_{\text{tan}} = r\,\alpha = 0.25\times 0 = 0\,\text{m\,/s}^2\] \[v = r\omega = 0.25\times 60 = 15\,\text{m\,/s}\quad\Rightarrow\quad a_{c}=\frac{v^2}{r}=\frac{15^2}{0.25}=900\,\text{m\,/s}^2\] |
At \(t=1\,\text{s}\) the graph is flat, so \(\alpha=0\Rightarrow a_{\text{tan}}=0.\) Centripetal acceleration depends on instantaneous speed: \(a_c = v^2/r.\) |
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When a fan is turned off, its angular speed decreases from \( 10 \) \( \text{rad/s} \) to \( 6.3 \) \( \text{rad/s} \) in \( 5.0 \) \( \text{s} \). What is the magnitude of the average angular acceleration of the fan?
A disk of radius 35 cm rotates at a constant angular velocity of 10 rad/s. How fast does a point on the rim of the disk travel (in m/s)?
An ice skater performs a pirouette (a fast spin) by pulling in his outstretched arms close to his body. What happens to his angular momentum about the axis of rotation?
The system in the Figure is in equilibrium. A concrete block of mass \(225 \, \text{kg}\) hangs from the end of a uniform strut whose mass is \(45.0 \, \text{kg}\).
A sphere of mass \( M \) and radius \( r \), and rotational inertia \( I \) is released from the top of an inclined plane of height \( h \). The surface has considerable friction. Using only the variables mentioned, derive an expression for the sphere’s center of mass velocity.
The angular velocity of an electric motor is \(\omega = \left(20 – \frac{1}{2} t^2 \right) \, \text{rad/s}\), where \(t\) is in seconds.
A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time?
A disk, a hoop, and a solid sphere are released at the same time at the top of an inclined plane. They are all uniform and roll without slipping. In what order do they reach the bottom?
\( \text{Solid sphere: } I = \frac{2}{5}mR^2, \quad \text{Solid disk: } I = \frac{1}{2}mR^2, \quad \text{Hoop: } I = mR^2 \)
Young David experimented with slings before tackling Goliath. He found that he could develop an angular speed of \( 8.0 \) \( \text{rev/s} \) in a sling \( 0.60 \) \( \text{m} \) long. If he increased the length to \( 0.90 \) \( \text{m} \), he could revolve the sling only \( 6.0 \) times per second.
A race car travels in a circular track of radius \( 200 \) \( \text{m} \). If the car moves with a constant speed of \( 80 \) \( \text{m/s} \),
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| Kinematics | Forces |
|---|---|
| \(\Delta x = v_i t + \frac{1}{2} at^2\) | \(F = ma\) |
| \(v = v_i + at\) | \(F_g = \frac{G m_1 m_2}{r^2}\) |
| \(v^2 = v_i^2 + 2a \Delta x\) | \(f = \mu N\) |
| \(\Delta x = \frac{v_i + v}{2} t\) | \(F_s =-kx\) |
| \(v^2 = v_f^2 \,-\, 2a \Delta x\) |
| Circular Motion | Energy |
|---|---|
| \(F_c = \frac{mv^2}{r}\) | \(KE = \frac{1}{2} mv^2\) |
| \(a_c = \frac{v^2}{r}\) | \(PE = mgh\) |
| \(T = 2\pi \sqrt{\frac{r}{g}}\) | \(KE_i + PE_i = KE_f + PE_f\) |
| \(W = Fd \cos\theta\) |
| Momentum | Torque and Rotations |
|---|---|
| \(p = mv\) | \(\tau = r \cdot F \cdot \sin(\theta)\) |
| \(J = \Delta p\) | \(I = \sum mr^2\) |
| \(p_i = p_f\) | \(L = I \cdot \omega\) |
| Simple Harmonic Motion | Fluids |
|---|---|
| \(F = -kx\) | \(P = \frac{F}{A}\) |
| \(T = 2\pi \sqrt{\frac{l}{g}}\) | \(P_{\text{total}} = P_{\text{atm}} + \rho gh\) |
| \(T = 2\pi \sqrt{\frac{m}{k}}\) | \(Q = Av\) |
| \(x(t) = A \cos(\omega t + \phi)\) | \(F_b = \rho V g\) |
| \(a = -\omega^2 x\) | \(A_1v_1 = A_2v_2\) |
| Constant | Description |
|---|---|
| [katex]g[/katex] | Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface |
| [katex]G[/katex] | Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex] |
| [katex]\mu_k[/katex] and [katex]\mu_s[/katex] | Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion. |
| [katex]k[/katex] | Spring constant, in [katex]\text{N/m}[/katex] |
| [katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex] | Mass of the Earth |
| [katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex] | Mass of the Moon |
| [katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex] | Mass of the Sun |
| Variable | SI Unit |
|---|---|
| [katex]s[/katex] (Displacement) | [katex]\text{meters (m)}[/katex] |
| [katex]v[/katex] (Velocity) | [katex]\text{meters per second (m/s)}[/katex] |
| [katex]a[/katex] (Acceleration) | [katex]\text{meters per second squared (m/s}^2\text{)}[/katex] |
| [katex]t[/katex] (Time) | [katex]\text{seconds (s)}[/katex] |
| [katex]m[/katex] (Mass) | [katex]\text{kilograms (kg)}[/katex] |
| Variable | Derived SI Unit |
|---|---|
| [katex]F[/katex] (Force) | [katex]\text{newtons (N)}[/katex] |
| [katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy) | [katex]\text{joules (J)}[/katex] |
| [katex]P[/katex] (Power) | [katex]\text{watts (W)}[/katex] |
| [katex]p[/katex] (Momentum) | [katex]\text{kilogram meters per second (kgm/s)}[/katex] |
| [katex]\omega[/katex] (Angular Velocity) | [katex]\text{radians per second (rad/s)}[/katex] |
| [katex]\tau[/katex] (Torque) | [katex]\text{newton meters (Nm)}[/katex] |
| [katex]I[/katex] (Moment of Inertia) | [katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex] |
| [katex]f[/katex] (Frequency) | [katex]\text{hertz (Hz)}[/katex] |
Metric Prefixes
Example of using unit analysis: Convert 5 kilometers to millimeters.
Start with the given measurement: [katex]\text{5 km}[/katex]
Use the conversion factors for kilometers to meters and meters to millimeters: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]
Perform the multiplication: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]
Simplify to get the final answer: [katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]
Prefix | Symbol | Power of Ten | Equivalent |
|---|---|---|---|
Pico- | p | [katex]10^{-12}[/katex] | |
Nano- | n | [katex]10^{-9}[/katex] | |
Micro- | µ | [katex]10^{-6}[/katex] | |
Milli- | m | [katex]10^{-3}[/katex] | |
Centi- | c | [katex]10^{-2}[/katex] | |
Deci- | d | [katex]10^{-1}[/katex] | |
(Base unit) | – | [katex]10^{0}[/katex] | |
Deca- or Deka- | da | [katex]10^{1}[/katex] | |
Hecto- | h | [katex]10^{2}[/katex] | |
Kilo- | k | [katex]10^{3}[/katex] | |
Mega- | M | [katex]10^{6}[/katex] | |
Giga- | G | [katex]10^{9}[/katex] | |
Tera- | T | [katex]10^{12}[/katex] |
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