| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \tau = FR | The torque (\tau) exerted on the cylinder is due to the force F applied at a radius R. The formula for torque is the force times the perpendicular distance (radius in this case) from the axis of rotation. |
| 2 | \tau = I\alpha | Newton’s second law for rotation states that the torque is equal to the moment of inertia (I) times the angular acceleration (\alpha). |
| 3 | I = \frac{1}{2}MR^2 | The moment of inertia for a solid cylinder about its axis is given by this formula, where M is the mass and R is the radius of the cylinder. |
| 4 | FR = \frac{1}{2}MR^2 \alpha | Substitute the moment of inertia of the cylinder into the torque equation. |
| 5 | \alpha = \frac{2F}{MR} | Solve for the angular acceleration (\alpha) by isolating \alpha on one side of the equation. |
| 6 | \omega^2 = \omega_0^2 + 2\alpha \theta | Use the kinematic equation for rotational motion to relate the angular displacement (\theta) to the final angular velocity (\omega). Here, \omega_0 (initial angular velocity) is zero as the cylinder starts from rest. |
| 7 | \omega^2 = 2\alpha \theta | Substitute \omega_0 = 0 into the equation because the cylinder starts from rest. |
| 8 | \omega = \sqrt{2\alpha \theta} = \sqrt{\frac{4F\theta}{MR}} | Substitute the value of \alpha from Step 5 into the equation to find \omega. |
| 9 | K = \frac{1}{2}I\omega^2 | The total kinetic energy (K) of the rotating cylinder is given by the formula for rotational kinetic energy, where I is the moment of inertia and \omega is the angular velocity. |
| 10 | K = \frac{1}{2} \times \frac{1}{2}MR^2 \times \left(\frac{4F\theta}{MR}\right) | Substitute the expressions for I and \omega into the kinetic energy formula. |
| 11 | K = \frac{F\theta R}{2} | Simplify the equation to get the final expression for the kinetic energy. |
| 12 | K = \frac{F\theta R}{2} | Conclude with the neat, simplified expression for the kinetic energy of the cylinder after it has rotated through an angle \theta. |
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A 2 kg model rocket is launched with a thrust force of 275 N and reaches a height of 90 m, moving at 150 m/s at its peak. What is the average air resistance force acting on the rocket during its ascent?
A rotating, rigid body makes 10 complete revolutions in 10 seconds. What is its average angular velocity?
A 100 kg person is riding a 10 kg bicycle up a 25° hill. The hill is long and the coefficient of static friction is 0.9. The person rides 10 m up the hill then takes a rest at the top. If she then starts from rest from the top of the hill and rolls down a distance of 7 m before squeezing hard on the brakes locking the wheels. How much work is done by friction to bring the bicycle to a full stop, knowing that the coefficient of kinetic friction is 0.65?
A planet of constant mass orbits the sun in an elliptical orbit. Neglecting any friction effects, what happens to the planet’s rotational kinetic energy about the sun’s center?
A spring launches a 4 kg block across a frictionless horizontal surface. The block then ascends a 30° incline with a kinetic friction coefficient of 0.25, stopping after 55 m on the incline. If the spring constant is 800 N/m, find the initial compression of the spring. Disregard friction while in contact with the spring.
K = \frac{F\theta R}{2}
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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_1m_2}{r^2} |
| a = \frac{\Delta v}{\Delta t} | f = \mu N |
| R = \frac{v_i^2 \sin(2\theta)}{g} |
| Circular Motion | Energy |
|---|---|
| F_c = \frac{mv^2}{r} | KE = \frac{1}{2} mv^2 |
| a_c = \frac{v^2}{r} | PE = mgh |
| KE_i + PE_i = KE_f + PE_f |
| Momentum | Torque and Rotations |
|---|---|
| p = m v | \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 |
|---|
| F = -k x |
| T = 2\pi \sqrt{\frac{l}{g}} |
| T = 2\pi \sqrt{\frac{m}{k}} |
| Constant | Description |
|---|---|
| g | Acceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface |
| G | Universal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2 |
| \mu_k and \mu_s | Coefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion. |
| k | Spring constant, in \text{N/m} |
| M_E = 5.972 \times 10^{24} , \text{kg} | Mass of the Earth |
| M_M = 7.348 \times 10^{22} , \text{kg} | Mass of the Moon |
| M_M = 1.989 \times 10^{30} , \text{kg} | Mass of the Sun |
| Variable | SI Unit |
|---|---|
| s (Displacement) | \text{meters (m)} |
| v (Velocity) | \text{meters per second (m/s)} |
| a (Acceleration) | \text{meters per second squared (m/s}^2\text{)} |
| t (Time) | \text{seconds (s)} |
| m (Mass) | \text{kilograms (kg)} |
| Variable | Derived SI Unit |
|---|---|
| F (Force) | \text{newtons (N)} |
| E, PE, KE (Energy, Potential Energy, Kinetic Energy) | \text{joules (J)} |
| P (Power) | \text{watts (W)} |
| p (Momentum) | \text{kilogram meters per second (kgm/s)} |
| \omega (Angular Velocity) | \text{radians per second (rad/s)} |
| \tau (Torque) | \text{newton meters (Nm)} |
| I (Moment of Inertia) | \text{kilogram meter squared (kgm}^2\text{)} |
| f (Frequency) | \text{hertz (Hz)} |
General Metric Conversion Chart
Example of using unit analysis: Convert 5 kilometers to millimeters.
Start with the given measurement: \text{5 km}
Use the conversion factors for kilometers to meters and meters to millimeters: \text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}
Perform the multiplication: \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}
Simplify to get the final answer: \boxed{5 \times 10^6 \, \text{mm}}
Prefix | Symbol | Power of Ten | Equivalent |
|---|---|---|---|
Pico- | p | 10^{-12} | |
Nano- | n | 10^{-9} | |
Micro- | µ | 10^{-6} | |
Milli- | m | 10^{-3} | |
Centi- | c | 10^{-2} | |
Deci- | d | 10^{-1} | |
(Base unit) | – | 10^{0} | |
Deca- or Deka- | da | 10^{1} | |
Hecto- | h | 10^{2} | |
Kilo- | k | 10^{3} | |
Mega- | M | 10^{6} | |
Giga- | G | 10^{9} | |
Tera- | T | 10^{12} |
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