PHY 151 Lecture 4 B 4 4 Particle

PHY 151: Lecture 4 B • 4. 4 Particle in Uniform Circular Motion • 4. 5 Relative Velocity

PHY 151: Lecture 4 B Motion in Two Dimensions 4. 4 Particle in Uniform Circular Motion

Accelerated Motion • No Acceleration ØObject is at rest ØObject is moving in a straight line with constant speed • Acceleration ØObject is moving in a straight line with changing speed ØObject is moving in a curve with constant speed ØObject is moving in a curve with changing speed

Uniform Circular Motion • Uniform circular motion occurs when an object moves in a circular path with a constant speed • The associated analysis model is a particle in uniform circular motion • An acceleration exists since the direction of the motion is changing – This change in velocity is related to an acceleration • The constant-magnitude velocity vector is always tangent to the path of the object

Changing Velocity in Uniform Circular Motion • The change in the velocity vector is due to the change in direction • The direction of the change in velocity is toward the center of the circle • The vector diagram shows

Centripetal Acceleration - 1 • The acceleration is always perpendicular to the path of the motion • The acceleration always points toward the center of the circle of motion • This acceleration is called the centripetal acceleration

Centripetal Acceleration - 2 • The magnitude of the centripetal acceleration vector is given by • The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

Period • The period, T, is the time required for one complete revolution • The speed of the particle would be the circumference of the circle of motion divided by the period • Therefore, the period is defined as

Period - Example • The tangential speed of a particle on a rotating wheel is 3. 0 m/s • Particle is 0. 20 m from the axis of rotation • How long will it take for the particle to go through one revolution? Ød = vt Ø 2 pr = v. T ØT = 2 pr/v = 2 p(0. 20)/3. 0 = 0. 42 s

Tangential Acceleration • The magnitude of the velocity could also be changing • In this case, there would be a tangential acceleration • The motion would be under the influence of both tangential and centripetal accelerations – Note the changing acceleration vectors

Total Acceleration • The tangential acceleration causes the change in the speed of the particle • The radial acceleration comes from a change in the direction of the velocity vector

Total Acceleration, equations • The tangential acceleration: • The radial acceleration: • The total acceleration: – Magnitude – Direction • Same as velocity vector if v is increasing, opposite if v is decreasing

PHY 151: Lecture 4 B Motion in Two Dimensions 4. 5 Relative Velocity

Relative Velocity • Two observers moving relative to each other generally do not agree on the outcome of an experiment • However, the observations seen by each are related to one another • A frame of reference can described by a Cartesian coordinate system for which an observer is at rest with respect to the origin

Different Measurements, example • Observer A measures point P at +5 m from the origin • Observer B measures point P at +10 m from the origin • The difference is due to the different frames of reference being used.

Different Measurements, another example • The man is walking on the moving beltway • The woman on the beltway sees the man walking at his normal walking speed • The stationary woman sees the man walking at a much higher speed. – The combination of the speed of the beltway and the walking • The difference is due to the relative velocity of their frames of reference

Relative Velocity, generalized • Reference frame SA is stationary • Reference frame SB is moving to the right relative to SA at – This also means that SA moves at – relative to SB • Define time t = 0 as that time when the origins coincide

Notation • The first subscript represents what is being observed • The second subscript represents who is doing the observing • Example – The velocity of B (and attached to frame SB) as measured by observer A

Relative Velocity, 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 – • is the velocity of the particle P measured by observer A • is the velocity of the particle P measured by observer B • These are called the Galilean transformation equations

Acceleration in Different Frames of Reference • The derivative of the velocity equation will give the acceleration equation • The acceleration of the particle measured by an observer in one frame of reference is the same as that measured by any other observer moving at a constant velocity relative to the first frame

Acceleration, cont. • Calculating the acceleration gives • Since • Therefore,