A 3300-m-high mountain is located on the equator. How much faster does a climber on top of the mountain move than a surfer at a nearby beach? The earth’s radius is 6400 km and the earth’s mass is 5.97 x 10^{24}.

**.01 m/s ^{2}** faster on the mountain than on the beach.

Note: Do not assume gravity to be 9.81 m/s^{2} on the beach.

Phy can also check your working. Just snap a picture!

- Statistics

Advanced

Mathematical

GQ

A spacecraft somewhere in between the earth and the moon experiences 0 net force acting on it. This is because the earth and the moon pull the spacecraft in equal but opposite directions. Find the distance *D* away from Earth, such that the spacecraft experiences zero net force. The distance between the Moon and Earth is ~3.844 x 10^{8} m.

NOTE: You may need the mass of the earth and moon. You can find this in the formula table.

- Gravitation, Linear Forces

Intermediate

Proportional Analysis

GQ

A child on Earth has a weight of 500N. Determine the weight of the child if the earth was to triple in both mass and radius (3M and 3r).

- Gravitation, Linear Forces

Advanced

Mathematical

GQ

A 2.2 \times 10^{21} \, \text{kg} moon orbits a distant planet in a circular orbit of radius 1.5 \times 10^8 \, \text{m}. It experiences a 1.1 \times 10^{19} \, \text{N} gravitational pull from the planet. What is the moon’s orbital period in earth days?

- Circular Motion, Gravitation

Intermediate

Conceptual

GQ

A satellite in circular orbit around the Earth moves at constant speed. This orbit is maintained by the force of gravity between the Earth and the satellite, yet no work is done on the satellite. How is this possible?

- Circular Motion, Energy, Gravitation

Intermediate

Proportional Analysis

GQ

The Earth’s radius is 6.37 x 10^{6} 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 m/s^{2}？

- Gravitation

Advanced

Mathematical

FRQ

The International Space Station has a mass of 4.2 x10^{5} kg and orbits Earth at a distance of 4.0 x10^{2} km above the surface. Earth has a radius of 6.37 x10^{6} m, and mass of 5.97 x10^{24} kg. Calculate the following:

- Circular Motion, Gravitation

Intermediate

Proportional Analysis

FRQ

Imagine a hypothetical planet that has two moons. Moon #1 is in a circular orbit of radius R and has a mass M.

- Gravitation

Intermediate

Mathematical

FRQ

A satellite circling Earth completes each orbit in 132 minutes.

- Circular Motion, Gravitation

Intermediate

Mathematical

GQ

Consider a neutron star with a mass equal to the sun, a radius of 10 km, and a rotation period of 1.0 s. What is the radius of a geosynchronous orbit about the neutron star? The mass of the sun can be found in the formula sheet above.

- Circular Motion, Gravitation

Advanced

Conceptual

GQ

The gravitational force that the moon exerts on Earth is often cited as the source for the tides we witness. However, the gravitational force the Sun exerts on Earth is over 100 times greater than the force the moon exerts on Earth.

Why is the force from the moon credited for the tides, and not the force from the sun?

- Gravitation

**.01 m/s ^{2}** faster on the mountain than on the beach.

Note: Do not assume gravity to be 9.81 m/s^{2} on the beach.

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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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