Representing Motion & Significant Figures

Counting Significant Figures

Significant figures are the digits in a measurement that carry meaning contributing to its precision. When performing calculations, answers should be reported with the correct number of significant figures.

• Key Point: When multiplying or dividing, the result should have as many significant figures as the measurement with the fewest significant figures.

• Example: Estimating gallons of diesel used in Arizona in one day using population, cars per person, mileage, and fuel efficiency.

Motion in One and Two Dimensions

Catch/Overtake Problems

These problems involve two objects moving such that one overtakes the other. The position and velocity equations for constant acceleration are used.

• Key Equation:

• Application: A van starts from rest with acceleration, while another moves at constant speed. Set their positions equal to solve for the time and speed at overtaking.

Position-Time Graphs

Position-time graphs show how an object's position changes over time. The slope of the graph at any point gives the velocity.

• Key Point: Velocity is zero where the tangent to the curve is horizontal.

• Example: Determining instants when velocity is zero by finding where the graph has horizontal tangents.

Projectile Motion

Projectile motion involves objects launched into the air, subject to gravity. The motion can be analyzed in horizontal and vertical components.

• Key Equations:

  • Horizontal distance:

  • Vertical motion:

• Example: Calculating the horizontal distance a rock travels when launched from a mountain edge.

Projectiles from Moving Vehicles

When a projectile is launched from a moving object, its initial velocity is the vector sum of the vehicle's velocity and the launch velocity.

• Key Point: Use vector addition to find the total initial velocity.

• Application: Calculating the distance a magic ball travels when thrown from a descending genie.

Forces & Newton's Laws

Vertical Forces & Acceleration

Newton's Second Law relates the net force on an object to its acceleration: .

• Key Point: When multiple objects are connected, analyze the system as a whole and consider tension forces.

• Example: Calculating tension in ropes pulling boats along a frictionless surface.

Systems of Objects with Friction

When objects are connected and friction is present, the net force includes frictional forces.

• Key Equation:

• Application: Calculating acceleration and tension in a system with a slab, box, and pulley.

Static Friction

Static friction prevents motion up to a maximum value: .

• Key Point: If the applied force is less than , the object does not move.

• Example: Determining the friction force on a salt block when pushed.

Equilibrium & Elasticity

Equilibrium in 2D

For an object in equilibrium, the sum of forces and torques must be zero.

• Key Equations:

• Example: Calculating tensions in cables holding a container at an angle.

Circular Motion, Orbits & Gravity

Acceleration Due to Gravity

The gravitational acceleration at the surface of a planet is given by:

• Application: Calculating for an exoplanet with different mass and radius than Earth.

Gravitational Forces in 2D

Escape velocity is the minimum speed needed for an object to escape a planet's gravitational field:

• Application: Calculating the maximum radius of a celestial object allowing escape by jumping.

Rotational Motion

Types of Acceleration in Rotation

Rotational motion involves angular and tangential acceleration.

• Key Equations:

  • Angular acceleration:

  • Tangential acceleration:

• Application: Calculating tangential acceleration for a toy car on a loop.

Net Torque & Sign of Torque

Torque causes rotational acceleration and is given by:

• Application: Calculating torque applied to a disk with known moment of inertia and angular acceleration.

Torque on Discs & Pulleys

Systems with pulleys and rotating masses require analysis of forces and torques.

• Key Point: The net torque determines the angular acceleration of the pulley.

• Example: Finding acceleration of elevator cars connected by a pulley.

Torque & Equilibrium

Levers and trolleys use torque to lift objects more easily.

• Key Equation:

• Application: Analyzing the forces that allow a worker to lift heavier loads using a trolley.

Momentum

Angular Momentum of a Point Mass

Angular momentum for a point mass is given by:

• Application: Calculating angular momentum for a block on a merry-go-round.

Intro to Angular Momentum

Angular momentum depends on the moment of inertia and angular velocity.

• Key Point: Units of angular momentum are or .

• Application: Determining units for coefficients in angular displacement equations.

Push-Away Problems (Conservation of Momentum)

When an object explodes or separates, conservation of momentum applies:

• Application: Calculating the speed and direction of fragments after a satellite explodes.

Energy & Work

Net Work & Work-Energy Theorem

The work-energy theorem states that the net work done on an object equals its change in kinetic energy:

• Application: Calculating the minimum static friction coefficient needed for a bicycle to move in a certain direction.

Work on Inclined Planes

Work done by various forces on an inclined plane can be calculated using:

• Application: Calculating work done by weight, normal force, and applied force as a crate moves up an incline.

Summary Table: Key Equations and Concepts

Additional info: These study notes cover representative problems from chapters on motion, forces, energy, rotation, and equilibrium, as found in a college-level Physics course. Each problem type is linked to a major concept or equation from the standard curriculum.