Before solving the question, we can find the radius of the ball using Pythagorean theorem to get .866 m. We can also use the trig to solve for the angle each rope makes with the horizontal (30° for both ropes).

**Sum of Forces in the Horizontal Direction:**

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

1 | \cos(30) = \frac{\sqrt{3}}{2} | Cosine of 30^\circ. |

2 | F_{\text{centripetal}} = \frac{mv^2}{r} | Centripetal force for circular motion. |

3 | T_1 \cos(\theta) + T_2 \cos(\theta) = \frac{mv^2}{r} | Sum of horizontal components of tension equals centripetal force. |

4 | T_1 \frac{\sqrt{3}}{2} + T_2 \frac{\sqrt{3}}{2} = \frac{(0.5)(7.2)^2}{0.866} | Substitute values for m, v, r, and \cos(\theta). |

5 | \frac{\sqrt{3}}{2}(T_1 + T_2) = 29.93 | Calculate centripetal force and factor out \frac{\sqrt{3}}{2}. |

**Sum of Forces in the Vertical Direction:**

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

1 | \sin(30) = \frac{1}{2} | Sine of 30^\circ. |

2 | w = mg | Weight of the sphere. |

3 | T_2 \sin(\theta) + mg – T_1 \sin(\theta) = 0 | Vertical forces must balance: upward tensions and downward weight. |

4 | T_2 \frac{1}{2} + (0.5)(9.8) – T_1 \frac{1}{2} = 0 | Substitute values for m, g, and \sin(\theta). |

5 | \frac{1}{2}(T_2 – T_1) + 4.9 = 0 | Factor out \frac{1}{2} and calculate weight. |

**Solving for Tensions:**

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

1 | Solve equations | Use the system of equations to solve for T_1 and T_2. |

2 | T_1 \approx 22.18 \text{ N} | Numerical solution for T_1. |

3 | T_2 \approx 12.38 \text{ N} | Numerical solution for T_2. |

Final Tensions:

- Upper wire: \boxed{T_1 \approx 22.18 \text{ N}}
- Lower wire: \boxed{T_2 \approx 12.38 \text{ N}}

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

- Statistics

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

Advanced

Proportional Analysis

GQ

Two identical satellites are placed in orbit of two different planets. Satellite A orbits Mars, and Satellite B orbits Jupiter. The orbital speeds of each satellite are the same. Which satellite has a greater orbital radius?

- Circular Motion, Gravitation

Advanced

Proportional Analysis

MCQ

A block starts at rest on a frictionless inclined track which then turns into a circular loop of radius *R* and is vertical. In terms of *R *and constants, find the minimum height *h *above the bottom of the loop the block must start from so it makes it around the loop.

- Circular Motion

Advanced

Proportional Analysis

MCQ

A 250 newton centripetal force acts on a car moving at a constant speed in a horizontal circle. If the same force is applied, but the radius is made smaller, what happens to the speed *v* and the frequency *f* of the car?

- Circular Motion

Intermediate

Mathematical

FRQ

A 2 kg ball is swung in a vertical circle. The length of the string the ball is attached to is 0.7 m. It takes 0.4 s for the ball to travel one revolution ( assume ball travels at constant speed).

- Circular Motion, Tension

Beginner

Mathematical

GQ

A 2.0 kg ball on the end of a 0.65 m long string is moving in a vertical circle. At the bottom of the circle, its speed is 4.0 m/s. Find the tension in the string.

- Circular Motion

Advanced

Mathematical

FRQ

A neighbor’s child wants to go to a carnival to experience the wild rides. The neighbor is worried about safety because one of the rides looks particularly dangerous. She knows that you have taken physics and so asks you for advice.

The ride in question has a 4 kg chair which hangs freely from a 10 m long chain attached to a pivot on the top of a tall tower. When the child enters the ride, the chain is hanging straight down. The child is then attached to the chair with a seat belt and shoulder harness. When the ride starts up, the chain rotates about the tower. Soon the chain reaches its maximum speed and remains rotating at that speed, which corresponds to one rotation about the tower every 3 seconds.

When you ask the operator, he says the ride is perfectly safe. He demonstrates this by sitting in the stationary chair. The chain creaks but holds, and he weighs 90 kg.

- Circular Motion

Advanced

Proportional Analysis

GQ

The distance from earth to sun is 1.0 AU. The distance from Saturn to sun is 9 AU. Find the period of Saturn’s orbit in years. You can assume that the orbits are circular.

- Circular Motion, Gravitation

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

Intermediate

Mathematical

FRQ

A satellite circling Earth completes each orbit in 132 minutes.

- Circular Motion, Gravitation

Upper wire: 22 N; Lower wire: 12 N

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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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