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| Derivation or Formula | Reasoning |
|---|---|
| \[L = I\omega\] | The total angular momentum of the man–platform system about the rotation axis is conserved because external torques are negligible; therefore \(L\) is constant even if \(I\) or \(\omega\) change. |
| \[I_0 = I_{\text{disk}} + mR^2\] | Initial moment of inertia when the man stands at the rim: the disk contributes \(I_{\text{disk}}\) and the man contributes \(mR^2\). |
| \[I’ = I_{\text{disk}} + mr^2,\quad \omega’ = \frac{I_0}{I’}\,\omega_0\] | If the man moves radially inward to radius \(r< R\), \(I’ < I_0\). Because \(L\) is constant, the angular speed becomes \(\omega’ > \omega_0\). Student A is therefore wrong that the platform slows; he is also wrong that \(L\propto I\)—it is \(L\)=constant. |
| \[L = I_{\text{disk}}\,\omega + m r^2(\omega – \Omega)\] | When the man runs clockwise with angular speed \(\Omega\) relative to the platform (opposite the platform’s counter-clockwise \(\omega\)), his angular momentum about the axis increases: \(m r^2(\omega – \Omega)\). Therefore his gain in angular momentum is balanced by opposite LOSS in the disk’s angular momentum. The platform’s \(\omega\) therefore decreases, in accord with Student B’s idea. |
| \[\tau_{\text{ext}} = 0\,\Rightarrow\,\dot L = 0\] | No external torque means the angular momentum of the entire system is unchanged, so the “man-platform system” as a whole does not experience an angular acceleration. Student B is mistaken on that point; only the individual parts (man vs. platform) accelerate relative to each other. |
Just ask: "Help me solve this problem."
| Wagon | Wheel Structure | Moment of Inertia | Wheel Mass | Wheel Radius |
|---|---|---|---|---|
| Wagon \(A\) | Solid disk | \[\frac{1}{2} M R^2\] | \[ 0.5 \, \text{kg} \] | \[ 0.1 \, \text{m} \] |
| Wagon \(B\) | Solid disk | \[\frac{1}{2} M R^2\] | \[ 0.2 \, \text{kg} \] | \[ 0.1 \, \text{m} \] |
| Wagon \(C\) | Hollow hoop | \[M R^2\] | \[ 0.1 \, \text{kg} \] | \[ 0.1 \, \text{m} \] |
Three wagons have identical total mass (including their wheels) and each has four wheels. However, the wheels on each wagon have different designs with varying mass distributions and radii as shown in a reference chart. When accelerating each wagon from a standstill to \( 10 \) \( \text{m/s} \), which wagon requires the most energy input?
The moment of inertia of a uniform solid sphere (mass \( M \), radius \( R \)) about a diameter is \( \frac{2}{5}MR^2 \). The sphere is placed on an inclined plane (angle \( \theta \)) and released from rest.
A force of 17 N is applied to the end of a 0.63-m long torque wrench at an angle 45° from a line joining the pivot point to the handle. What is the magnitude of the torque about the pivot point produced by this force?
Which of the following must be zero if an object is spinning at a constant rate? There may be more than one right answer.
Consider a uniform hoop of radius \( R \) and mass \( M \) rolling without slipping. Which is larger, its translational kinetic energy or its rotational kinetic energy?
Student A is correct that moving toward the center decreases the system’s moment of inertia, but incorrectly concludes that this causes angular momentum to decrease. In reality, angular momentum is conserved in the absence of external torques, so the platform’s angular velocity must increase when the man moves inward. Student B correctly notes that running in the opposite direction can apply a frictional torque, which redistributes angular momentum within the system, causing the platform to slow down. However, they overlook that the total angular momentum of the system remains conserved, even though the platform’s rotation rate can change.
Some key points to include:
State that angular momentum is conserved if no external torque is acting.
Explain that moving inward decreases moment of inertia, so angular velocity increases to conserve angular momentum.
Running in the opposite direction of rotation applies a frictional torque that can slow the platform.
Emphasize that any changes are internal redistributions; total angular momentum remains constant.
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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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