# Part (a) Free Body Diagram Explanation

Please visualize or draw out the FBD as described:

– **Gravity ( \vec{mg} )**: Acts directly downwards. With a mass of m = 90 + 12 = 102 kg.

– **Normal Force ( \vec{N} )**: Acts perpendicular to the surface of the incline.

– **Frictional Force ( \vec{f} )**: Acts parallel to the incline against the direction of motion during ascent and in the opposite direction of velocity during descent.

– **Component of Gravitational Force down the incline ( \vec{mg} \sin(\theta) )**: Helps in descending and resists during ascending. Here \theta = 30^\circ .

# Part (b) Calculation of Work Done by Friction to Stop the Bicycle

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

1 | F_f = \mu_k N | Frictional force, F_f , is the product of the coefficient of kinetic friction, \mu_k , and the normal force, N . |

2 | N = mg \cos(\theta) | Normal force is the component of the gravitational force perpendicular to the incline. |

3 | F_f = 0.7 \times 102 \times 9.8 \times \cos(30^\circ) | Calculate F_f using \mu_k = 0.7 , m = 102 kg, g = 9.8 \, \text{m/s}^2 , and \theta = 30^\circ . |

4 | W_f = -F_f d | Work done by friction, W_f , is the product of the frictional force and the distance, d , over which it acts, with a negative sign indicating work done against the motion. |

5 | W_f = -0.7 \times 102 \times 9.8 \times \cos(30^\circ) \times 9 | Substitute values to calculate the work done. The distance d = 9 m. |

6 | W_f \approx -5454 \, \text{J} |
Calculated work done by friction; it’s negative as it opposes the direction of motion. |

# Part (c) Explanation – Difficulty of Traveling Up vs. Down the Incline

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

1 | F_{\text{gravity, down}} = mg \sin(\theta) | Component of gravitational force along the incline that assists in descending and opposes during ascending. |

2 | F_{\text{friction, up}} = \mu_s N | Static friction opposing the upward motion, which is higher due to increased force requirements. |

3 | F_{\text{net, up}} = F_{\text{gravity, down}} + F_{\text{friction, up}} | Sum of forces opposing the ascent, both the gravitational pull back down the incline and the frictional force. |

4 | F_{\text{net, down}} = F_{\text{gravity, down}} – F_{\text{friction, down}} | Net force during descent is reduced because friction (now kinetic) is less than static friction and gravity assists in motion. |

5 | Comparison | The cyclist finds it harder to travel up due to higher net opposing force (more friction and gravity opposing motion). |

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

- Statistics

Advanced

Mathematical

FRQ

A 20 g piece of clay moving at a speed of 50 m/s strikes a 500 g pendulum bob at rest. The length of a string is 0.8 m. After the collision the clay-bob system starts to oscillate as a simple pendulum.

- Energy, Momentum, Pendulums, Simple Harmonic Motion

Intermediate

Conceptual

MCQ

A rubber ball bounces off of a wall with an initial speed v and reverses its direction so its speed is v right after the bounce. As a result of this bounce, which of the following quantities of the ball are conserved?

- Energy, Momentum

Advanced

Mathematical

GQ

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?

- Energy

Advanced

Mathematical

MCQ

A simple pendulum consists of a sphere tied to the end of a string of negligible mass. The sphere is pulled back until the string is horizontal and then released from rest. Assume the gravitational potential energy is zero when the sphere is at its lowest point.

What angle will the string make with the horizontal when the kinetic energy and the potential energy of the sphere-Earth system are equal?

- Energy

Intermediate

Proportional Analysis

MCQ

If you want to double the momentum of a gas molecule, by what factor must you increase its kinetic energy?

- Energy, Momentum

- FBD with 3 forces.
- W_f \approx -5454 \, \text{J}
- The cyclist finds it harder to travel up due to higher net opposing force (more friction and gravity opposing motion).

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!

The most advanced version of Phy. Currently 50% off, for early supporters.

per month

Billed Monthly. Cancel Anytime.

Trial –> Phy Pro

- Unlimited Messages
- Unlimited Image Uploads
- Unlimited Smart Actions
- Unlimited UBQ Credits
- 30 --> 300 Word Input
- 3 --> 15 MB Image Size Limit
- 1 --> 3 Images per Message
- 200% Memory Boost
- 150% Better than GPT
- 75% More Accurate, 50% Faster
- Mobile Snaps
- Focus Mode
- No Ads

A quick explanation

UBQ credits are specifically used to grade your FRQs and GQs.

You can still view questions and see answers without credits.

Submitting an answer counts as 1 attempt.

Seeing answer or explanation counts as a failed attempt.

Lastly, check your average score, across every attempt, in the top left.

MCQs are 1 point each. GQs are 1 point. FRQs will state points for each part.

Phy can give partial credit for GQs & FRQs.

Phy sees everything.

It customizes responses, explanations, and feedback based on what you struggle with. Try your best on every question!

Understand you mistakes quicker.

For GQs and FRQs, Phy provides brief feedback as to how you can improve your answer.

Aim to increase your understadning and average score with every attempt!

10 Free Credits To Get You Started

*Phy Pro members get unlimited credits