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| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[\vec{F}_g = m_c\,\vec{g}\] | Weight acts downward; here \(m_c = 900\,\text{kg}\). |
| 2 | \[\vec{N}= -\vec{F}_g\] | Normal force from bed acts upward, balancing weight (level road, no vertical acceleration). |
| 3 | \[\vec{f}_s \text{ (leftward)}\] | Static friction on crate points left, supplying the horizontal deceleration so that the crate does not slide. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[\Delta x = \frac{v_i + v_x}{2}\,\Delta t\] | Constant–acceleration displacement relation. |
| 2 | \[55 = \frac{25 + v_x}{2}\,(3.0)\] | Substitute \(\Delta x = 55\,\text{m},\ v_i = 25\,\text{m/s},\ \Delta t = 3.0\,\text{s}.\) |
| 3 | \[v_x = 11.7\,\text{m/s}\] | Solve for final speed after braking distance. |
| 4 | \[v_x = v_i + a\,\Delta t\] | Linear velocity–time equation. |
| 5 | \[11.7 = 25 + a(3.0)\] | Insert known values. |
| 6 | \[a = -4.4\,\text{m/s}^2\] | Negative sign shows acceleration is opposite the motion (deceleration). |
| 7 | \[\boxed{\,|a| = 4.4\,\text{m/s}^2\,}\] | Requested magnitude. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[\vec{f}_s = m_c\,|a|\] | Static friction provides the horizontal net force \(m_c a\). |
| 2 | \[f_{s,\max}= \mu_s N = \mu_s m_c g\] | Maximum static friction formula with \(N = m_c g\). |
| 3 | \[\mu_s m_c g \ge m_c |a|\] | No sliding requires static friction capability to exceed demand. |
| 4 | \[\mu_{\min}= \frac{|a|}{g}= \frac{4.4}{9.8}=0.45\] | Mass cancels; compute numerical value. |
| 5 | \[\boxed{\,\mu_{\min}=0.45\;\text{(static)}\,}\] | The friction is static because no relative motion occurs. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[a = \frac{v_x – v_i}{\Delta t}= \frac{25 – 0}{10}=2.5\,\text{m/s}^2\] | Uniform acceleration from rest to \(25\,\text{m/s}\) in \(10\,\text{s}\). |
| 2 | \[F_s = m_c a\] | Spring force supplies crate’s required horizontal force. |
| 3 | \[F_s = kx\] | Hooke’s law for spring with constant \(k = 9200\,\text{N/m}\). |
| 4 | \[x = \frac{m_c a}{k}= \frac{900(2.5)}{9200}=0.24\,\text{m}\] | Solve for extension. |
| 5 | \[\boxed{\,x = 0.24\,\text{m}\,}\] | Numerical answer for maximum extension during acceleration. |
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \[a = 0\] | Constant speed implies zero acceleration. |
| 2 | \[F_s = m_c a = 0\Rightarrow x = 0\] | No net horizontal force required, so spring force and thus extension become zero. |
| 3 | \[\boxed{\,x_{\text{const}} < x_{\text{accel}}\,}\] | Extension is less (indeed zero) compared with part (d) when acceleration was present. |
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A communications satellite orbits the Earth at an altitude of \(35{,}000 \, \text{km}\) above the Earth’s surface. Take the mass of Earth to be \(6 \times 10^{24} \, \text{kg}\) and the radius of Earth to be \(6.4 \times 10^6 \, \text{m}\). What is the satellite’s velocity?
A runner pushes against the track to sprint forward. Which two action–reaction FORCE pairs are involved? Select two letters.
A train consists of \(50\) cars, each of which has a mass of \(6.1 \times 10^{3} \, \text{kg}\). The train has an acceleration of \(8.0 \times 10^{-2} \, \text{m/s}^2\). Ignore friction and determine the tension in the coupling at the following places:
The figure shows a truck pulling three crates across a rough road. Which of the following best describes the directions of all the horizontal forces acting on crate 2?
The Earth’s radius is \(6.37 \times 10^{6} \, \text{m}\). What is the radius of a planet that has the same mass as Earth but on which the free-fall acceleration is \(5.50 \, \text{m/s}^2\)?
A pulley system consists of two blocks of mass \( 5 \) \( \text{kg} \) and \( 10 \) \( \text{kg} \), connected by a rope of negligible mass that passes over a pulley of radius \( 0.1 \) \( \text{m} \) and mass \( 2 \) \( \text{kg} \). The pulley is free to rotate about its axis. The system is released from rest, and the block of mass \( 10 \) \( \text{kg} \) starts to move downwards. Assume the pulley has a frictional force of \(5.7\) Newtons acting on the outer edge of the pulley.
A block sliding down an frictionless inclined plane is experiencing both gravitational and normal forces; which force’s magnitude changes when the angle of the incline is increased?
A runner is moving at \( 4 \) \( \text{m/s} \). She is opposed by magic in the form of air resistance, which exerts a constant \( 20 \) \( \text{Newtons} \) force in the direction opposite her velocity. At what rate is she using energy to remain at constant velocity?
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 conical pendulum is formed by attaching a ball of mass \( m \) to a string of length \( \ell \), then allowing the ball to move in a horizontal circle of radius \( r \). The following figure shows that the string traces out the surface of a cone, hence the name.
\(4.4\,\text{m/s}^2\)
\(\mu_{\min}=0.45,\,\text{static}\)
\(0.24\,\text{m}\)
\(\text{less than}\)
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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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