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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | [katex] U_{\text{spring}} = \frac{1}{2} k x^2 [/katex] | The potential energy stored in the compressed spring, where [katex] k [/katex] is the spring constant and [katex] x [/katex] is the compression distance. |
| 2 | [katex] W_{\text{friction}} = \mu_k \cdot m \cdot g \cdot \cos(\theta) \cdot d [/katex] | Work done against friction while moving up the incline: [katex] \mu_k [/katex] is the coefficient of kinetic friction, [katex] m [/katex] is mass, [katex] g [/katex] is acceleration due to gravity, [katex] \theta [/katex] is the angle of incline, and [katex] d [/katex] is the distance traveled up the incline. |
| 3 | [katex] PE = m \cdot g \cdot h [/katex] | The gravitational potential energy gained by the block at the end of the incline. Here, [katex] h [/katex] is the vertical height, calculated further in Step 5. |
| 4 | [katex] U_{\text{spring}} = W_{\text{friction}} + PE [/katex] | Conservation of mechanical energy: The total mechanical energy (spring potential energy) initially equals the sum of the work done against friction and the gravitational potential energy at the end. |
| 5 | [katex] h = d \cdot \sin(\theta) [/katex] | Using the distance [katex] d [/katex] traveled up the incline and the angle [katex] \theta [/katex] to calculate the height [katex] h [/katex]. |
| 6 | Substituting [katex] h [/katex], [katex] W_{\text{friction}} [/katex], and [katex] PE [/katex] in step 4: | [katex] \frac{1}{2} k x^2 = \mu_k \cdot m \cdot g \cdot \cos(\theta) \cdot d + m \cdot g \cdot d \cdot \sin(\theta) [/katex] |
| 7 | Simplifying and solving for [katex] x [/katex]: | [katex] x^2 = \frac{2(\mu_k \cdot m \cdot g \cdot \cos(\theta) \cdot d + m \cdot g \cdot d \cdot \sin(\theta))}{k} [/katex] [katex] x = \sqrt{\frac{2d(mg(\mu_k \cdot \cos(\theta) + \sin(\theta)))}{k}} [/katex] |
| 8 | [katex] x = \sqrt{\frac{2 \cdot 55 \cdot (4 \cdot 9.8 \cdot (0.25 \cdot \cos(30^\circ) + \sin(30^\circ)))}{800}} [/katex] | Substitute values for [katex] d = 55 \, \text{m} [/katex], [katex] m = 4 \, \text{kg} [/katex], [katex] g = 9.8 \, \text{m/s}^2 [/katex], [katex] \mu_k = 0.25 [/katex], [katex] \theta = 30^\circ [/katex], and [katex] k = 800 \, \text{N/m} [/katex] to find [katex] x [/katex]. |
| 9 | [katex] x = \boxed{1.965\, m} [/katex] | Final value. |
Just ask: "Help me solve this problem."
A mass moving with a constant speed \( u \) encounters a rough surface and comes to a stop. The mass takes a time \( t \) to stop after encountering the rough surface. The coefficient of dynamic friction between the rough surface and the mass is \( 0.40 \). Which of the following expressions gives the initial speed \( u \)?
An object at rest suddenly explodes into two fragments (m1 and m2) by an explosion. Fragment m1 acquires 3 times the kinetic energy of the other. What is the ratio of m1 to m2?
A 0.2 kg object is attached to a horizontal spring undergoes SHM with the total energy of 0.4 J. The kinetic energy as a function of position presented by the graph.
Friction provides the force needed for a car to travel around a flat, circular race track. Answer the following:
A forward horizontal force of 12 N is used to pull a 240 N crate at constant velocity across a horizontal floor. The coefficient of friction is
[katex] x = \boxed{1.965\, m} [/katex]
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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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