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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| Part (a): Determine the final angular speed when power returns. | ||
| 1 | \( v_i = 500\,\text{rpm} \times \frac{2\pi\,\text{rad}}{60\,\text{s}} = \frac{500\times 2\pi}{60} = \frac{50\pi}{3}\,\text{rad/s} \) | Convert the initial speed from revolutions per minute to radians per second. |
| 2 | \( \Delta \theta = 200\,\text{rev} \times 2\pi = 400\pi\,\text{rad} \) | Calculate the total angular displacement during the 30 s power outage (each revolution is \(2\pi\) rad). |
| 3 | \( \Delta \theta = \frac{1}{2}(v_i + v_x)\,t \) | Use the kinematic equation for constant angular acceleration relating displacement, initial and final speeds over time \(t = 30\,s\). |
| 4 | \( v_x = \frac{2\Delta \theta}{t} – v_i = \frac{2(400\pi)}{30} – \frac{50\pi}{3} = \frac{800\pi}{30} – \frac{50\pi}{3} \) | Solve for the final angular speed \(v_x\) after 30 s. |
| 5 | \( \frac{800\pi}{30} = \frac{80\pi}{3}, \quad v_x = \frac{80\pi}{3} – \frac{50\pi}{3} = \frac{30\pi}{3} = 10\pi\,\text{rad/s} \) | Simplify the expression to obtain \(v_x\). This is equivalent to \(10\pi\,rad/s\) which can be converted to \(300\,rpm\) if desired. |
| 6 | \( \boxed{v_x = 10\pi\,\text{rad/s}} \) | Final answer for part (a): the flywheel spins at \(10\pi\,rad/s\) when power returns. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| Part (b): Determine the time to stop and total revolutions if power did not return. | ||
| 1 | \( \alpha = \frac{v_x – v_i}{t} = \frac{10\pi – \frac{50\pi}{3}}{30} = \frac{\frac{30\pi – 50\pi}{3}}{30} = -\frac{20\pi}{90} = -\frac{2\pi}{9}\,\text{rad/s}^2 \) | Determine the constant angular deceleration \(\alpha\) using the change in angular velocity over 30 s. |
| 2 | \( 0 = v_i + \alpha T_{\text{stop}} \quad \Rightarrow \quad T_{\text{stop}} = -\frac{v_i}{\alpha} \) | Set the final angular velocity to zero to solve for the total stopping time \(T_{\text{stop}}\) from the moment of power failure. |
| 3 | \( T_{\text{stop}} = -\frac{\frac{50\pi}{3}}{-\frac{2\pi}{9}} = \frac{50\pi}{3} \times \frac{9}{2\pi} = 75\,\text{s} \) | Simplify to find that the flywheel takes 75 s to come to a complete stop. |
| 4 | \( \Delta \theta_{\text{total}} = \frac{v_i + 0}{2}T_{\text{stop}} = \frac{\frac{50\pi}{3}}{2} \times 75 = \frac{50\pi\times75}{6} = 625\pi\,\text{rad} \) | Calculate the total angular displacement using the average angular speed during deceleration. |
| 5 | \( \text{Revolutions} = \frac{625\pi}{2\pi} = 312.5\,\text{rev} \) | Convert radians to revolutions since \(2\pi\) rad correspond to one complete revolution. |
| 6 | \( \boxed{T_{\text{stop}} = 75\,\text{s} \quad \text{and} \quad \text{Total Revolutions} = 312.5\,\text{rev}} \) | Final answers for part (b): the flywheel stops in 75 s making a total of 312.5 revolutions. |
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A horizontal uniform meter stick of mass 0.2 kg is supported at its midpoint by a pivot point. A mass of 0.1 kg is attached to the left end of the meter stick, and another mass of 0.15 kg is attached to the right end of the meter stick. The meter stick is free to rotate in the horizontal plane around the pivot point. What is the tension in the string supporting the left end of the meter stick?
A solid sphere of mass [katex] 1.5 \, \text{kg} [/katex] and radius [katex] 15 \, \text{cm} [/katex] rolls without slipping down a [katex] 35^\circ[/katex] incline that is [katex] 7 \, \text{m} [/katex] long. Assume it started from rest. The moment of inertia of a sphere is [katex] I= \frac{2}{5}MR^2 [/katex].
A wheel 31 cm in diameter accelerates uniformly from 240rpm to 360rpm in 6.8 s. How far will a point on the edge of the wheel have traveled in this time?
The angular velocity of a rotating disk of radius 20 cm increases from 1 rad/s to 3 rad/s in 0.5 s. What is the linear tangential acceleration of a point on the rim of the disk during this time interval?
Part (a): \(\boxed{10\pi\,\text{rad/s}}\) (which is equivalent to 300 rpm).\nPart (b): \(\boxed{75\,\text{s}}\) to come to a complete stop with a total of \(\boxed{312.5\,\text{rev}}\).
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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] |
General Metric Conversion Chart
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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