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Part A – Velocity of block
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( \frac{1}{2} m v_x^2 = m g (3.0) \) | Apply conservation of energy. The gravitational potential energy lost by the block when dropped from \(3.0\) m is converted entirely into kinetic energy just before impact. |
| 2 | \( \frac{1}{2} v_x^2 = g (3.0) \) | Cancel the mass \(m\) (since it appears on both sides) to isolate the velocity term. |
| 3 | \( v_x^2 = 2 g (3.0) \) | Multiply both sides by 2 to solve for \(v_x^2\). |
| 4 | \( v_x = \sqrt{2 \times 9.8 \times 3.0} \) | Substitute \(g = 9.8\ \text{m/s}^2\) into the equation. |
| 5 | \( v_x = \sqrt{58.8} \approx 7.67\ \text{m/s} \) | Calculate the square root to determine the final speed just before the spring is struck. |
| 6 | \( \boxed{v_x \approx 7.67\ \text{m/s}} \) | This is the final answer for part (a). |
Part B – Spring Energy
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( \Delta h = 3.0\,\text{m} + 0.50\,\text{m} = 3.5\,\text{m} \) | The block drops 3.0 m before striking the spring and an additional \(0.50\,\text{m}\) while compressing the spring, so the total vertical drop is \(3.5\,\text{m}\). |
| 2 | \( U_{\text{spring}} = m g \Delta h \) | The total elastic potential energy stored in the spring equals the gravitational potential energy lost by the block over the total drop \(\Delta h\). |
| 3 | \( U_{\text{spring}} = 240 \times 9.8 \times 3.5 \) | Substitute \(m = 240\,\text{kg}\), \(g = 9.8\,\text{m/s}^2\), and \(\Delta h = 3.5\,\text{m}\) into the energy formula. |
| 4 | \( U_{\text{spring}} \approx 8232\,\text{J} \) | Performing the multiplication, \(240 \times 9.8 = 2352\) and \(2352 \times 3.5 \approx 8232\), yielding the stored elastic energy. |
| 5 | \( \boxed{U_{\text{spring}} \approx 8232\,\text{J}} \) | This is the final answer for part (b): the total elastic potential energy stored in the spring. |
Part C – Spring Constant
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( \frac{1}{2} k (\Delta x)^2 = U_{\text{spring}} \) | The elastic potential energy stored in a compressed spring is given by \(\frac{1}{2} k (\Delta x)^2\), where \(\Delta x\) is the compression distance. |
| 2 | \( \frac{1}{2} k (0.50)^2 = 8232 \) | Substitute \(\Delta x = 0.50\,\text{m}\) and \(U_{\text{spring}} = 8232\,\text{J}\) into the formula. |
| 3 | \( \frac{1}{2} k \times 0.25 = 8232 \) | Calculate \((0.50)^2 = 0.25\) and rewrite the equation. |
| 4 | \( 0.125 k = 8232 \) | Simplify \(\frac{1}{2} \times 0.25\) to \(0.125\). |
| 5 | \( k = \frac{8232}{0.125} = 8232 \times 8 \) | Solve for \(k\) by dividing both sides by \(0.125\); note that \(\frac{1}{0.125} = 8\). |
| 6 | \( k \approx 65856\,\text{N/m} \) | Multiply to find the spring constant. |
| 7 | \( \boxed{k \approx 6.59 \times 10^4\,\text{N/m}} \) | This is the final answer for part (c), rounded to three significant figures. |
Just ask: "Help me solve this problem."
A crate rests on a horizontal surface and a woman pulls on it with a 10-N force. No matter what the orientation of the force, the crate does not move. From least to greatest, rank the normal force on the crate.
A ski tow carries people to the top of a nearby mountain. It operates on a slope of angle \( 15.7^\circ \) of length \( 260 \) \( \text{m} \). The rope moves at a speed of \( 13.0 \) \( \text{km/h} \) and provides power for \( 54 \) riders at one time, with an average mass per rider of \( 67.0 \) \( \text{kg} \).
If the coefficient of static friction is \( \mu_s = 0.5 \), how much force must be applied to a spring (spring constant of \( 0.8 \) \( \text{N/m} \)) which is attached to a block of wood (mass \( 4.0 \) \( \text{kg} \)) in order to just begin to move the block?
Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the Earth? Which accelerates more?
A rubber ball bounces off of a wall with an initial speed v and reverses its direction so its speed is v right after the bounce. As a result of this bounce, which of the following quantities of the ball are conserved?
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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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