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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[\text{Coordinate System: } +x \text{ (East), } +y \text{ (North)}\] | Define the coordinate system with East as the positive \(x\) direction and North as the positive \(y\) direction. |
| 2 | \[F_{1x} = 170 \;\text{N}, \quad F_{1y} = 0 \;\text{N}\] | Student 1 pulls Eastward with \(170\,\text{N}\); hence, all force is in the \(x\) direction. |
| 3 | \[F_{2x} = 0 \;\text{N}, \quad F_{2y} = -100 \;\text{N}\] | Student 2 pulls Southward with \(100\,\text{N}\); therefore, the \(y\) component is negative. |
| 4 | \[F_{3x} = -200\sin(20^\circ), \quad F_{3y} = 200\cos(20^\circ)\] | Student 3 pulls with \(200\,\text{N}\) at \(20^\circ\) west of north. The \(y\) component is \(200\cos(20^\circ)\) (northward) and the \(x\) component is \(-200\sin(20^\circ)\) (westward). |
| 5 | \[F_{\text{net},x} = 170 – 200\sin(20^\circ)\] | Sum the \(x\) components: Student 1 contributes \(170\,\text{N}\) east, and Student 3 contributes \(-200\sin(20^\circ)\,\text{N}\) (west). |
| 6 | \[F_{\text{net},y} = -100 + 200\cos(20^\circ)\] | Sum the \(y\) components: Student 2 gives \(-100\,\text{N}\) (south) and Student 3 gives \(200\cos(20^\circ)\,\text{N}\) (north). |
| 7 | \(200\sin(20^\circ) \approx 68.4 \;\text{N}, \quad 200\cos(20^\circ) \approx 187.9 \;\text{N}\) | Calculate the approximate numerical values of the components for Student 3. |
| 8 | \[F_{\text{net},x} \approx 170 – 68.4 = 101.6 \;\text{N}\] | Compute the net \(x\) component using the approximated value. |
| 9 | \[F_{\text{net},y} \approx -100 + 187.9 = 87.9 \;\text{N}\] | Compute the net \(y\) component using the approximated value. |
| 10 | \[F_{\text{net}} = \sqrt{(101.6)^2 + (87.9)^2} \approx 134.4 \;\text{N}\] | Find the magnitude of the net force using the Pythagorean theorem. |
| 11 | \[\theta = \tan^{-1}\left(\frac{87.9}{101.6}\right) \approx 40.9^\circ\]\] | Determine the direction of the net force measured as the angle north of east. |
| 12 | \[\boxed{134.4 \;\text{N},\; 40.9^\circ \; \text{north of east}}\] | State the final net force magnitude and its direction. |
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Three blocks of masses \(m_3 = 1.0 \, \text{kg}\), \(m_2 = 2.0 \, \text{kg}\), and \(m_1 = 4.0 \, \text{kg}\) are connected by massless strings, one of which passes over a frictionless pulley of negligible mass, as shown above.
A westward–moving car is changing its speed. The net force on the car ____.
A \(2.2 \times 10^{21} \, \text{kg}\) moon orbits a distant planet in a circular orbit of radius \(1.5 \times 10^8 \, \text{m}\). It experiences a \(1.1 \times 10^{19} \, \text{N}\) gravitational pull from the planet. What is the moon’s orbital period in Earth days?
The distance from earth to sun is \(1 \, \text{AU}\). The distance from Saturn to sun is \(9 \, \text{AU}\). Find the period of Saturn’s orbit in years. You can assume that the orbits are circular.
During lunch, Alex and Jordan argue about inertia. Alex says if she spins a basketball faster, it will have greater inertia. Jordan argues that inertia only depends on the ball’s mass, not its speed. Who is correct?
An object has a mass of 10 kg. For each case below answer the questions and provide an example.
Friction provides the force needed for a car to travel around a flat, circular race track. Answer the following:
A constant force of 8.0 N is exerted on a 16 kg object initially at rest. How much speed will the object gain after 4 seconds?
A person is trying to judge whether a picture (mass = 1.42 kg) is properly positioned by temporarily pressing it against a wall. The pressing force is perpendicular to the wall. The coefficient of static friction between the picture and the wall is 0.62. What is the minimum amount of pressing force that must be used?
Determine the force needed to push a \( 150 \) \( \text{kg} \) body up a smooth \( 30^\circ \) incline with an acceleration of \( 6 \) \( \text{m/s}^2 \).
\(\boxed{134.4\,\text{N}\text{ at }40.9^\circ\text{ north of east}}\)
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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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