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To solve the problem using principles of work and energy, we consider the system comprising the two masses. We know that one mass will move up while the other moves down, and we’re given the displacement [katex] d = 0.8 [/katex] meters for each mass.
Assuming the 12 kg mass moves downward and the 9 kg mass moves upward, we need to find the common velocity [katex] v [/katex] after they have moved 0.8 meters, taking into account that energy is conserved in this isolated system (ignoring air resistance and friction). Here’s the step-by-step analysis:
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | [katex] m_1 = 12 \, \text{kg} [/katex] [katex] m_2 = 9 \, \text{kg} [/katex] |
Define the masses where [katex] m_1 [/katex] is the 12 kg mass and [katex] m_2 [/katex] is the 9 kg mass. |
| 2 | [katex] g = 9.81 \, \text{m/s}^2 [/katex] | Acceleration due to gravity is [katex] g [/katex]. |
| 3 | [katex] \Delta h = 0.8 \, \text{m} [/katex] | Both masses displace by 0.8 m, one moving up and the other down. |
| 4 | [katex] U_{\text{initial}} = 0 \, \text{J} [/katex] | Initial potential energy is set to zero for simplicity as we’re only interested in changes. |
| 5 | [katex] U_{\text{final}} = -m_1 g \Delta h + m_2 g \Delta h [/katex] | Final potential energy obtained by considering the gain in height by [katex] m_2 [/katex] and loss in height by [katex] m_1 [/katex]. |
| 6 | [katex] U_{\text{final}} = -12 \times 9.81 \times 0.8 + 9 \times 9.81 \times 0.8 [/katex] | Substitute the known values to find [katex] U_{\text{final}} [/katex]. |
| 7 | [katex] U_{\text{final}} = -23.544 \, \text{J} [/katex] | Calculate the final potential energy. |
| 8 | [katex] KE_{\text{initial}} = 0 \, \text{J} [/katex] | Initial kinetic energy, assuming they start from rest. |
| 9 | [katex] KE_{\text{final}} = \frac{1}{2} (m_1 + m_2) v^2 [/katex] | Kinetic energy for the system of the two moving masses at the final state. |
| 10 | [katex] KE_{\text{final}} = \frac{1}{2} (12+9) v^2 [/katex] | Expression for the final kinetic energy of both masses. |
| 11 | [katex] KE_{\text{final}} = 10.5 v^2 [/katex] | Simplifying the expression for final kinetic energy. |
| 12 | [katex] KE_{\text{final}} = -U_{\text{final}} [/katex] | By conservation of energy, change in potential energy equals change in kinetic energy. |
| 13 | [katex] 10.5 v^2 = 23.544 [/katex] | Substitute computed value of [katex] U_{\text{final}} [/katex]. |
| 14 | [katex] v^2 = \frac{23.544}{10.5} [/katex] | Solve for [katex] v^2 [/katex]. |
| 15 | [katex] v = \sqrt{\frac{23.544}{10.5}} [/katex] | Calculate [katex] v [/katex]. |
| 16 | [katex] v \approx 1.49 \, \text{m/s} [/katex] | Final velocity of the masses after traveling 0.8 m. |
This result indicates the common speed of both masses after they have moved 0.8 meters, based on the conservation of mechanical energy.
Just ask: "Help me solve this problem."
An apple is released from rest 500 m above the ground. Due to the combined forces of air resistance and gravity, it has a speed of 40 m/s when it reaches the ground. What percentage of the initial mechanical energy of the apple-Earth system was dissipated due to air resistance? Take the potential energy of the apple-Earth system to be zero when the apple reaches the ground.
If you want to double the momentum of a gas molecule, by what factor must you increase its kinetic energy?
A ball of radius \( r \) rolls on the inside of a circular track of radius \( R \). If the ball starts from rest at the left vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping?
A lighter car and a heavier truck, each traveling to the right with the same speed [katex] v [/katex] hit their brakes. The retarding frictional force F on both cars turns out to be constant and the same. After both vehicles travel a distance [katex] D [/katex] (and both are still moving), which of the following statements is true?
Jill does twice as much work as Jack does and in half the time. Jill’s power output is
1.49 m/s
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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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