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

1 | v_{\text{critical}} = \sqrt{gR} | The critical speed v_{\text{critical}} is the minimum speed at the top of the loop which allows the roller coaster to remain in contact with the track and complete the loop without falling off. It is found by equating the gravitational force to the required centripetal force for circular motion at the top of the loop. |

2 | v = 2v_{\text{critical}} = 2\sqrt{gR} | It is given that the roller coaster crosses at twice the critical speed. Therefore, the actual speed v is twice the critical speed. |

3 | F_{\text{net}} = m \frac{v^2}{R} | The net force needed for circular motion at the top of the loop is given by the centripetal force formula where m is the mass of the roller coaster, v is the velocity, and R is the radius of the loop. |

4 | F_{\text{net}} = m \frac{(2\sqrt{gR})^2}{R} = 4mg | Substitute v = 2\sqrt{gR} into the centripetal force formula and simplify. |

5 | F_{\text{net}} = F_{\text{n}} + mg = 4mg | The net force F_{\text{net}} at the top of the loop is the sum of the normal force F_{\text{n}} exerted by the track on the roller coaster and the gravitational force mg . |

6 | F_{\text{n}} + mg = 4mg | Setting the net inward force (which includes the normal and gravitational forces) equal to the required centripetal force for motion at this speed. |

7 | F_{\text{n}} = 4mg \,-\, mg = 3mg | Solving for F_{\text{n}} by subtracting mg from each side. |

8 | \frac{F_{\text{n}}}{F_{\text{g}}} = \frac{3mg}{mg} = 3 | The ratio of the normal force to the gravitational force is \frac{F_{\text{n}}}{F_{\text{g}}} . Substituting the values from the previous step. |

9 | \frac{F_{\text{n}}}{F_{\text{g}}} = 3 | The final answer, indicating the ratio is 3, matching option (b). |

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

- Statistics

Advanced

Mathematical

GQ

Two wires are tied to the 500 g sphere shown below. The sphere revolves in a horizontal circle at a constant speed of 7.2 m/s. What is the tension in the upper wire? What is the tension in the lower wire?

- Circular Motion

Intermediate

Mathematical

GQ

A communications satellite orbits the Earth at an altitude of 35,000 km above the Earth’s surface. Take the mass of Earth to be 6 \times 10^{24} \text{ kg} the the radius of Earth to be 6.4 \times 10^6 \text{ m}. What is the satellite’s velocity?

- Centripetal Acceleration, Circular Motion, Gravitation, Linear Forces

Advanced

Conceptual

MCQ

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?

- Angular Momentum, Kepler's Law, Rotational Energy

Beginner

Mathematical

GQ

A car travels at a constant speed around a circular track whose radius is 2.6 km. the car goes once around the track in 360 s. What is the magnitude of the centripetal acceleration of the car?

- Circular Motion

Advanced

Mathematical

GQ

Riders in a carnival ride stand with their backs against the wall of a circular room of diameter 8.0 m. The room is spinning horizontally about an axis through its center at a rate of 45 rev/min when the floor drops so that it no longer provides any support for the riders. What is the *minimum* coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall?

- Circular Motion

Intermediate

Conceptual

MCQ

A compressed spring mounted on a disk can project a small ball. When the disk is not rotating, as shown in the top view above, the ball moves radially outward. The disk then rotates in a counterclockwise direction as seen from above, and the ball is projected outward at the instant the disk is in the position shown above. Which of the following best shows the subsequent path of the ball relative to the ground?

- Circular Motion

Advanced

Mathematical

GQ

An Olympic bobsled team goes through a horizontal curve at a speed of 120 km/hr. If the radius of curvature is 10.0 m, what is the apparent weight the crew experiences-express in terms of mg.

- Circular Motion

Intermediate

Conceptual

MCQ

A delivery truck is traveling north. It then turns along a leftward circular curve. This the packages in the truck to slide to the RIGHT. Which of the following is true of the net force on the packages as they are sliding?

- Circular Motion

Intermediate

Mathematical

FRQ

A discus is held at the end of an arm that starts at rest. The average angular acceleration of 54 \, \text{rad/s}^2 lasts for 0.25 s. The path is circular and has radius 1.1 m.

Note: A discuss is a heavy, flattened circular object for throwing.

- Centripetal Acceleration, Circular Motion, Rotational Kinematics

Intermediate

Mathematical

FRQ

In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we can approximate it as a 50 km diameter circular orbit with a period of 11 days.

- Circular Motion, Gravitation

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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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