Physics 1501: Lecture 6 Today’s Agenda Announcements: Introduction of your Lab TA’s Physics Learning Center P201: Schedule posted on the door Homeworks HW 01: Due this Friday Lowest HW will be dropped Masteringphysics: still a few kink HW 02: due next Monday 1

Today’s topics Circular motion Introduction to Newton’s laws

Recap: x = R cos()= R cos(t) y = R sin()= R sin(t)  = tan-1 (y/x) R v s t (x,y)  = t s = v t s = R = Rt v = R

Centripetal Acceleration UCM results in acceleration: Magnitude: a = v2 / R =  R Direction: - r (toward center of circle) ^ a R 

Radial and Tangential Quantities What about non-uniform circular motion ? v a aq is along the direction of motion ar is perpendicular to the direction of motion

Lecture 6, ACT 1 The Pendulum q = 30° Which statement best describes the motion of the pendulum bob at the instant of time drawn ? the bob is at the top of its swing. which quantities are non-zero ? 1m A) vr = 0 ar = 0 vq  0 aq  0 B) Vr = 0 ar  0 vq  0 aq = 0 C) vr = 0 ar  0 vq = 0 aq  0

Example: Newton & the Moon What is the acceleration of the Moon due to its motion around the earth? What we know (Newton knew this also): T = 27.3 days = 2.36 x 106 s (period ~ 1 month) R = 3.84 x 108 m (distance to moon) RE = 6.35 x 106 m (radius of earth) R RE

Moon... Calculate angular frequency: So  = 2.66 x 10-6 s-1. Now calculate the acceleration. a = 2R = 0.00272 m/s2 = .000278 g direction of a is toward center of earth (-r ). ^

Moon... amoon g This inspired him to propose that FMm  1 / R2 So we find that amoon / g = .000278 Newton noticed that RE2 / R2moon = .000273 R RE amoon g This inspired him to propose that FMm  1 / R2 (more on gravity later)

Inertial Reference Frames: A Reference Frame is the place you measure from. It’s where you nail down your (x,y,z) axes! An Inertial Reference Frame (IRF) is one that is not accelerating. We will consider only IRF’s in this course. Valid IRF’s can have fixed velocities with respect to each other. More about this later when we discuss forces. For now, just remember that we can make measurements from different vantage points.

Lecture 6 – ACT 2 Relative Motion Consider an airplane flying on a windy day. A pilot wants to fly from New Haven to Bradley airport. Having asked a friendly physics student, she knows that Bradley is 120 miles due north of New Haven and there is a wind blowing due east at 30 mph. She takes off from New Haven Airport at noon. Her plane has a compass and an air-speed indicator to help her navigate. She uses her compass at the start to aim her plane north, and her air speed indicator tells her she is traveling at 120 mph with respect to the air. After one hour, She is at Bradley She is due east of Bradley She is southeast of Bradley

Lecture 6, ACT 3 Relative Motion You are swimming across a 50m wide river in which the current moves at 1 m/s with respect to the shore. Your swimming speed is 2 m/s with respect to the water. You swim across in such a way that your path is a straight perpendicular line across the river. How many seconds does it take you to get across? a) b) c) d) 2m/s 1m/s 50m

Dynamics Isaac Newton (1642 - 1727) published Principia Mathematica in 1687. In this work, he proposed three “laws” of motion: Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. Law 2: For any object, FNET = F = ma Law 3: Forces occur in pairs: FA ,B = - FB ,A (For every action there is an equal and opposite reaction.)

(See Newton’s Second Law) Force We have an idea of what a force is from everyday life. Physicist must be precise. A force is that which causes a body to accelerate. (See Newton’s Second Law) Examples Contact Forces Field Forces (Non-Contact) (physical contact (action through between objects) “empty” space) Kicking a ball Moon and Earth On a microscopic level, all forces are non-contact

(See Newton’s First Law) Mass We have an idea of what mass is from everyday life. Physicist must be precise. mass (for this class) is a quantity that specifies how much inertia an object has. (See Newton’s First Law) Mass is an inherent property of an object. Mass and weight are different quantities. weight is a force.

Newton’s First Law An object subject to no external forces moves with a constant velocity if viewed from an inertial reference frame. If no forces act, there is no acceleration. The above statement can be thought of as the definition of inertial reference frames. An IRF is a reference frame that is not accelerating (or rotating) with respect to the “fixed stars”. If one IRF exists, infinitely many exist since they are related by any arbitrary constant velocity vector!

Is Storrs a good IRF? Is Storrs accelerating? YES! Storrs is on the Earth. The Earth is rotating. What is the centripetal acceleration of Storrs? T = 1 day = 8.64 x 104 sec, R ~ RE = 6.4 x 106 meters . Plug this in: aS = .034 m/s2 ( ~ 1/300 g) Close enough to 0 that we will ignore it. Storrs is a pretty good IRF.

Newton’s Second Law The acceleration of an object is directly proportional to the net force acting upon it. The constant of proportionality is the mass. Units The units of force are kg m/s2 = Newtons (N) The English unit of force is Pounds (lbs)

Lecture 6, ACT 4 Newton’s Second Law A constant force is exerted on a cart that is initially at rest on an air table. The force acts for a short period of time and gives the cart a certain final speed. Force Cart Air Track For a second shot, I can apply a force only half as large (I’m getting tired). To reach the same final speed, for how long must I apply the force ? A) 4 x as long B) 2 x as long C) Same time D) 1/2 as long E) 1/4 x as long

Lecture 6, ACT 5 Newton’s Second Law I push with a force of 2 Newtons on a cart that is initially at rest on an air table with no air. I push for a second. Because there is no air, the cart stops after I finish pushing. It has traveled a certain distance. Force Cart Air Track For a second shot, I push just as hard but keep pushing for 2 seconds. The distance the cart moves the second time versus the first is, A) 4 x as long B) 2 x as long C) Same D) 1/2 as long E) 1/4 x as long

Lecture 6, ACT 6 Newton’s Second Law A constant force is exerted on a cart that is initially at rest on an air table. This force is applied for a short period of time and the cart acquires a certain final speed, which I call vf1. Force Cart Air Track I repeat the experiment, but this time the cart is already moving with constant speed when I start applying the force. After exerting the same constant force for the same time interval, the cart’s final speed is, A) vf1 B) 2vf1 C) vf12 D) cannot be determined from the information given.

Newton’s Third Law: If object 1 exerts a force on object 2 (F1,2 ) then object 2 exerts an equal and opposite force on object 1 (F2,1) F1,2 = -F2,1 For every “action” there is an equal and opposite “reaction” This is among the most abused and misunderstood concepts in physics, along with Einstein’s ideas of relative motion (inertial reference frames) and Heisenberg’s uncertainty principle.