Physics 1A Lecture 3B
Review of Last Lecture For constant acceleration, motion along different axes act independently from each other (independent kinematic equations) One is free to choose a coordinate system that simplifies these equations (e.g., aligned with acceleration vector) Time of flight depends on maximum height of object, maximum for vertical trajectory Range depends on horizontal velocity and time of flight, maximum for 45º angle Circular motion is an accelerated motion due to change in velocity vector direction (even for constant speed)
Uniform circular motion R A special type of motion is circular motion at constant speed measured in radians spin rate (radians/second)
Uniform circular motion Is this accelerated motion? R YES! Acceleration is a change in velocity with respect to time change can be in magnitude or direction In this case it is a change in direction (magnitude = const.)
Acceleration and directional change of motion v 0 aδt : length of v changes by aδt (speed up/slow down) v
Acceleration and directional change of motion v v 0 aδt : length of v changes by aδt (speed up/slow down) : direction of v changes (if Δt -> dt small, no speed change)
Centripetal acceleration R In constant circular motion the acceleration is inward toward center of circle => centripetal acceleration What is?
Derive equation by dimensional analysis! Things we know: [v] = L T -1 [R] = L Thing we want: [a c ]= L T -2 Set up power-law equation: a c = v A R B L T -2 = L A T -A L B length dimension: 1 = A+B time dimension: -2 = -A => A = 2, B = -1 => a c = v 2 /R
Centripetal acceleration R In constant circular motion the acceleration is inward toward center of circle = centripetal acceleration What is?
Example: earth orbiting the sun E Period T time interval for one complete revolution: S Let s calculate the value of centripetal acceleration:
Tangential and radial acceleration A more general situation Velocity is not constant (both magnitude and direction) Curvature is not constant (radius R depends on position/time)
Tangential and radial acceleration The velocity is always tangent to the path The acceleration is at some angle to the path At each instant, motion can be approximated as circular The radius of that circular path is the radius of curvature of the path at that instant Total acceleration can be represented as a sum of two components (radial and tangential):
Tangential and radial acceleration Tangential component: rate of change in speed of the particle Radial component is a result of the change in direction of the velocity vector of the particle Total acceleration is a sum of two components:
Relative velocity Two observers moving relative to each other generally do not agree on the outcome of an experiment For example, the observer on the side of the road observes a different speed for the red car than does the observer in the blue car
Relative velocity, generalized Reference frame S is stationary Reference frame S is moving Define time t = 0 as that time when the origins coincide
Galilean transformation equations The positions as seen from the two reference frames are related through the velocity The derivative of the position equation will give the velocity equation This can also be expressed in terms of the observer O
Relative velocity: example Observer O is standing; observer O is in the blue car Both observers are measuring the speed of the red car, which is located at point P
Galilean transformation equations Although observers in two reference frames measure different velocities for the particle, they measure the same acceleration (if their relative velocity is constant) Hence: Such frames are called inertial reference frames
For Next Time (FNT) Check your Quiz 1 scores online Quiz 2 will cover Chapters 3 and 4 Start Reading Chapter 4 Finish the homework for Chapter 3