Ch4c

Chapter 4 Overview

• Subject: Physics 110

• Date: September 27

• Instructor: Dr. Reid

Key Concepts Discussed in Today's Lecture

• Circular motion

• Radial and Tangential acceleration

• Relationship to curved trajectories

• Caution regarding signs, as they are complex

• Relative motion

Analysis Model: Particle in Uniform Circular Motion

• Top view representation showing the motion of the particle

• The equation for uniform circular motion:

  • $rac{dv}{dt}$ indicates change in velocity over time

Detailed Analysis Model Equations

• The relationships are defined as follows:

  • $extbf{v} = extbf{r} rac{D extbf{v}}{D extbf{r}} = rac{D extbf{v}}{D extbf{r}} extbf{r}$

  • $a = D extbf{v} = rac{D extbf{v}}{D extbf{t}} extbf{r}$

• This model illustrates how particles behave under circular motion conditions, highlighting both linear and centripetal (radial) acceleration.

Centripetal Acceleration

• Formula for centripetal acceleration:

  • $a_c = rac{v^2}{r}$

• Where:

  • $v$ = speed of the particle

  • $r$ = radius of the circular motion

Relationships and Equations for Uniform Circular Motion

• The relationships incorporating period $T$ and angular velocity $extbf{w}$ include:

  • $v = 2 ext{π} r / T$

  • Substituting gives the equation:

  • $extbf{w} = rac{2 ext{π}}{T}$

• Deriving the radial acceleration gives us relationships:

  • $a = r extbf{w}^2$

  • $a = rac{v^2}{r}$

Example: Centripetal Acceleration of Earth

• To find the centripetal acceleration of the Earth:

  • Use the formula:

  • $a_c = rac{4 ext{π}^2 r}{T^2}$; where:

  • $r = 1.496 imes 10^{11} ext{ m}$ (average distance from Earth to Sun)

  • $T = 1 ext{ year} = 3.156 imes 10^7 ext{ s}$

  • Plugging the values into the formula gives the centripetal acceleration.

Tangential and Radial Acceleration

• The overall acceleration of the particle, given as:

  • $extbf{a} = extbf{a}<em>r + extbf{a}</em>t$

  • where

  • $extbf{a}_r = - rac{v^2}{r}$ (radial)

  • $extbf{a}_t = rac{dv}{dt}$ (tangential)

Quick Quiz Part I: Acceleration and Velocity Vectors

• When are the acceleration and velocity vectors parallel?

  • (a) when the path is circular

  • (b) when the path is straight

  • (c) when the path is a parabola

  • (d) never

Quick Quiz Part II: Perpendicular Vectors

• When are the acceleration and velocity vectors perpendicular everywhere along the path?

  • (a) when the path is circular

  • (b) when the path is straight

  • (c) when the path is a parabola

  • (d) never

Example: Accelerating Ball in Vertical Circle

• Description of scenario: A ball swings counterclockwise in a vertical circle at the end of a 1.50m long rope.

  • Given: At a position of 36.9° past the lowest point, total acceleration is $(-22.5 + 20.2 ext{ j}) ext{ m/s}^2$.

  • Tasks:

  • (a) Sketch a vector diagram showing the components of acceleration.

  • (b) Determine the magnitude of radial acceleration.

  • (c) Determine the speed and velocity of the ball.

Importance of Relative Motion

• Why relative motion is critical:

  • Observers in different frames of reference can measure different things, e.g., throwing a ball in the air.

  • Einstein’s theory of relativity states that physics must be consistent across all frames.

  • The aim is to understand how a constant relative velocity should be accounted for using Galilean transformations.

  • These transformations are specifically relevant for inertial frames of reference where no acceleration is present.

Example: Mythbusters Experiment

• Hypothetical analysis: A cannon fired at 50 mph from a truck moving at 50 mph illustrates how measurements vary in different frames of reference.

• The video included an animation to clarify the concept that observers must agree on the laws of physics regardless of their movement.

Relative Velocity and Relative Acceleration

• Explanation: A woman on a moving beltway observes a man at a slower speed than a stationary observer does.

Relative Velocity Equation

• Relative velocity can be described by the equation:

  • $extbf{v}<em>{AB} = extbf{v}</em>A + extbf{v}_B$

• Implication: This equation indicates how the perceived speeds of objects change depending on the observer's frame.

Galilean Transformation Equations

• Relationship between positions and velocities in inertial frames can be expressed as:

  • $rac{d}{dt} extbf{P} = rac{d}{dt} extbf{A} + rac{d}{dt} extbf{B}$

Relative Acceleration Equation

• The acceleration relationship:

  • $extbf{a}<em>{AB} = extbf{a}</em>A - extbf{a}_B$

• Indicates how accelerations perceived change between different frames of reference as well.

Example: Relative Velocity of Raindrops

• Scenario: A car moving east at 50.0 km/h and rain falling vertically, causing traces on the side windows to form an angle of 60.0° with the vertical.

• Tasks:

  • (a) Find the velocity of the rain with respect to the car.

  • (b) Find the velocity of the rain with respect to the Earth.

Example: Projectile Motion from Different Reference Frames

• Scenario: A student on a flatcar moving at 10.0 m/s throws a ball at a 60.0° angle horizontally, while an observer on the ground sees it rise vertically.

• Question: How high does the observer perceive the ball to rise?

Example: A Boat Crossing a River

• Scenario: A river flows at 0.500 m/s. A student swims upstream, travels 1.00 km, and returns.

• Tasks:

  • (a) Determine the trip time with the current.

  • (b) Determine time required in still water.

  • (c) Discuss reasons for increased time against the current.