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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[ F_\text{output} = mg \] | Calculate the force due to the mass of the rock. Here, \( m = 55.2 \, \text{kg} \) and \( g = 9.81 \, \text{m/s}^2 \). |
| 2 | \[ F_\text{output} = 55.2 \times 9.81 \] | Substitute values to find the force on the output plunger. |
| 3 | \[ F_\text{output} = 541.212 \, \text{N} \] | The force exerted by the rock on the output plunger. |
| 4 | \[ \frac{F_\text{input}}{A_\text{input}} = \frac{F_\text{output}}{A_\text{output}} \] | Use Pascal’s principle, which states that pressure is transmitted undiminished in an enclosed static fluid. |
| 5 | \[ F_\text{input} = \frac{F_\text{output} \times A_\text{input}}{A_\text{output}} \] | Rearrange to solve for the input force needed for equilibrium. |
| 6 | \[ F_\text{input} = \frac{541.212 \times 15}{65} \] | Substitute the area values: \( A_\text{input} = 15 \, \text{cm}^2 \) and \( A_\text{output} = 65 \, \text{cm}^2 \). |
| 7 | \[ F_\text{input} = 124.843 \, \text{N} \] | Calculate the force exerted on the input piston necessary for equilibrium. |
| 8 | \[ 124.843 = k_s \Delta x \] | Relate the input force to the spring constant \( k_s = 1250 \, \text{N/m} \) and the compression \( \Delta x \). |
| 9 | \[ \Delta x = \frac{124.843}{1250} \] | Solve for the compression of the spring. |
| 10 | \[ \Delta x = 0.0999 \, \text{m} \] | Convert the compression to meters. |
| 11 | \[ \boxed{9.99 \, \text{cm}} \] | Convert to centimeters and box the final answer. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[ A_\text{input} \Delta y_\text{input} = A_\text{output} \Delta y_\text{output} \] | Use the principle of conservation of volume in the hydraulic system. |
| 2 | \[ 15 \times 22.0 = 65 \times \Delta y_\text{output} \] | Substitute \( \Delta y_\text{input} = 22.0 \, \text{cm} \) and the areas. |
| 3 | \[ 330 = 65 \times \Delta y_\text{output} \] | Calculate the product of the input area and the distance. |
| 4 | \[ \Delta y_\text{output} = \frac{330}{65} \] | Solve for the rise in the output plunger’s height. |
| 5 | \[ \Delta y_\text{output} = 5.077 \, \text{cm} \] | The final rise in the output plunger. |
| 6 | \[ \boxed{5.08 \, \text{cm}} \] | Box the final answer after rounding to two decimal places. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[ P = P_0 + \rho g h \] | The absolute pressure at a depth \( h \) is given by this equation, where \( P_0 \) is atmospheric pressure. |
| 2 | \[ P = 101325 + 1000 \times 9.81 \times 0.85 \] | Substitute \( P_0 = 101325 \, \text{Pa} \), \( \rho = 1000 \, \text{kg/m}^3 \), \( g = 9.81 \, \text{m/s}^2 \), and the height \( h = 0.85 \, \text{m} \). |
| 3 | \[ P = 101325 + 8338.5 \] | Calculate the pressure contribution from the water column. |
| 4 | \[ P = 109663.5 \, \text{Pa} \] | Calculate the total absolute pressure at the bottom of the chamber. |
| 5 | \[ \boxed{109664 \, \text{Pa}} \] | Box the final answer after rounding to the nearest Pascal. |
Just ask: "Help me solve this problem."
Ben’s favorite ride at the Barrel-O-Fun Amusement Park is the Flying Umbrella, which is lifted by a hydraulic jack. The operator activates the ride by applying a force of \( 72 \) \( \text{N} \) to a \( 3.0 \) \( \text{cm} \) wide cylindrical piston, which holds the \( 20,000 \) \( \text{N} \) ride off the ground. What is the diameter of the piston that holds the ride?
A cylindrical tank of water (height \( H \)) is punctured at a height \( h \) above the bottom. How far from the base of the tank will the water stream land (in terms of \( h \) and \( H \))? What must the value of \( h \) be such that the distance at which the stream lands will be equal to \( H \)?
A cube of unknown material and uniform density floats in a container of water with \(60\%\) of its volume submerged. If this same cube were placed in a container of oil with density \(800\) \(\text{kg/m}^3\), what portion of the cube’s volume would be submerged while floating?
Why do you float higher in salt water than in fresh water?
How large must a heating duct be if air moving \( 3 \ \frac{\text{m}}{\text{s}} \) along it can replenish the air in a room of \( 300 \ \text{m}^3 \) volume every \( 15 \) minutes? Assume the air’s density 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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