| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[m_{\text{child+chair}}=25\,\text{kg}+4\,\text{kg}=29\,\text{kg}\] | The chain supports both the child and the chair during the ride, so the rotating mass is the combined mass. |
| 2 | \[m_{\text{operator+chair}}=90\,\text{kg}+4\,\text{kg}=94\,\text{kg}\] | The operator’s stationary demonstration also includes the chair, so the test load is the combined mass of the operator and chair. |
| 3 | \[T_{\text{test}}=(94)(9.81)\approx 922\,\text{N}\] | When the operator sits motionless, the chain tension is just the weight of the operator plus chair. This shows the chain can hold at least about \(922\,\text{N}\), not necessarily that this is its breaking force. |
| 4 | \[\omega=\frac{2\pi}{P}=\frac{2\pi}{3}\,\text{rad/s}\] | The chair completes one rotation every \(3\,\text{s}\), so its angular speed is \(2\pi/3\,\text{rad/s}\). |
| 5 | \[r=L\sin\theta\] | The ride is a conical pendulum. The chair moves in a horizontal circle whose radius is not the full chain length, but the horizontal component of the chain length. |
| 6 | \[T_{\text{ride}}\cos\theta=mg\] | The vertical component of the chain tension balances the weight of the child plus chair. |
| 7 | \[T_{\text{ride}}\sin\theta=m\omega^2r=m\omega^2L\sin\theta\] | The horizontal component of the tension provides the centripetal force needed for circular motion. |
| 8 | \[T_{\text{ride}}=m\omega^2L\] | Since the chair is rotating with \(\sin\theta\neq 0\), cancel \(\sin\theta\) from both sides of the horizontal force equation. |
| 9 | \[T_{\text{ride}}=(29)\left(\frac{2\pi}{3}\right)^2(10)\approx 1.27\times 10^3\,\text{N}\] | The tension in the chain during the ride with the child is about \(1270\,\text{N}\). |
| 10 | \[\cos\theta=\frac{mg}{T_{\text{ride}}}=\frac{g}{\omega^2L}\approx \frac{9.81}{\left(\frac{2\pi}{3}\right)^2(10)}\approx 0.224\] | This gives the angle of the chain from the vertical and confirms the conical-pendulum geometry. |
| 11 | \[\theta\approx \cos^{-1}(0.224)\approx 77.1^\circ\] | The chain is at a large angle from vertical when the ride rotates at the stated speed. |
| 12 | \[T_{\text{ride}}\approx 1270\,\text{N}>T_{\text{test}}\approx 922\,\text{N}\] | The moving child plus chair requires a larger chain tension than the load demonstrated by the stationary operator plus chair. |
| 13 | \[\boxed{\text{No, the operator has not shown that the ride is safe for the child.}}\] | The demonstration only proves the chain held a stationary load of about \(922\,\text{N}\). The ride requires about \(1270\,\text{N}\) while rotating, so the operator’s test is insufficient evidence of safety. |
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\(\boxed{\text{No, the operator has not shown that the ride is safe for a }25\,\text{kg}\text{ child.}}}\)
\(\boxed{T_{\text{test}}\approx 922\,\text{N},\quad T_{\text{ride}}\approx 1.27\times 10^3\,\text{N}}}\)
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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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