Step | Derivation/Formula | Reasoning |
---|---|---|

1 | m = 5.5\,\text{kg} | The mass of the block. |

2 | g = 9.81\,\text{m/s}^2 | Acceleration due to gravity. |

3 | \theta = 30^\circ | Angle of the incline. |

4 | \mu = 0.20 | Coefficient of friction between the block and the incline. |

5 | F_{\text{gravity}} = mg \sin(\theta) | The component of gravitational force pulling the block down the incline. |

6 | F_{\text{friction}} = \mu N | The frictional force acting opposite to the direction of motion, where N is the normal force. |

7 | N = mg \cos(\theta) | Normal force exerted by the incline on the block, perpendicular to the surface. |

8 | F_{\text{friction}} = \mu mg \cos(\theta) | Substituting the expression for normal force into the friction formula. |

9 | F_{\text{net}} = mg \sin(\theta) – \mu mg \cos(\theta) | The net force acting on the block along the incline, which is the gravitational force minus the frictional force. |

10 | a = \frac{F_{\text{net}}}{m} | Net acceleration of the block, calculated using Newton’s second law (F = ma). |

11 | a = g (\sin(\theta) – \mu \cos(\theta)) | Solving for a by substituting the expressions into acceleration formula. |

12 | a = 9.81 (\sin(30^\circ) – 0.20 \cos(30^\circ)) | Calculating the numerical value of acceleration by plugging in values for g, \theta, and \mu. |

13 | a = 9.81 (0.5 – 0.20 \times 0.866) | Using trigonometric values for sine and cosine of 30 degrees. |

14 | a = 9.81 \times (0.5 – 0.1732) | Continuing the calculation. |

15 | a = 9.81 \times 0.3268 \approx 3.21\,\text{m/s}^2 | Final calculation of the acceleration. |

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- Statistics

Beginner

Mathematical

GQ

What force would have to be applied to start a 12.3 kg wood block moving on a surface with a static coefficient of friction of 0.438?

- Linear Forces

Intermediate

Mathematical

GQ

Determine the force needed to push a 150 kg body up a smooth 30° incline with an acceleration of 6 m/s^{2}.

- Inclines, Linear Forces

Advanced

Mathematical

MCQ

Block m_{1} is stacked on top of block m2. Block m_{2} is connected by a light cord to block m_{3}, which is pulled along a frictionless surface with a force F as shown in the diagram above. Block m_{1} is accelerated at the same rate as block m_{2} because of the frictional forces between the two blocks. If all three blocks have the same mass m, what is the minimum coefficient of static friction between block m_{1} and block m_{2}?

- Linear Forces, Multi-Body Systems

Intermediate

Mathematical

MCQ

A small sphere hangs from a string attached to the ceiling of a uniformly accelerating train car. It is observed that the string makes an angle of 37° with respect to the vertical. The magnitude of the acceleration a of the train car is most nearly:

- Linear Forces

Advanced

Mathematical

GQ

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?

- Energy, Inclines

3.2 m/s^{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

Conversion Example

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} | 0.000000000001 |

Nano- | n | 10^{-9} | 0.000000001 |

Micro- | µ | 10^{-6} | 0.000001 |

Milli- | m | 10^{-3} | 0.001 |

Centi- | c | 10^{-2} | 0.01 |

Deci- | d | 10^{-1} | 0.1 |

(Base unit) | – | 10^{0} | 1 |

Deca- or Deka- | da | 10^{1} | 10 |

Hecto- | h | 10^{2} | 100 |

Kilo- | k | 10^{3} | 1,000 |

Mega- | M | 10^{6} | 1,000,000 |

Giga- | G | 10^{9} | 1,000,000,000 |

Tera- | T | 10^{12} | 1,000,000,000,000 |

- Some answers may be slightly off by 1% depending on rounding, etc.
- Answers will use different values of gravity. Some answers use 9.81 m/s
^{2}, and other 10 m/s^{2 }for calculations. - Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
- Bookmark questions that you can’t solve so you can come back to them later.
- Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!

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