Last Time: Start Rotational Motion (now thru mid Nov) Basics: Angular Speed, Angular Acceleration

Transcription

1 Last Time: Start Rotational Motion (now thru mid No) Basics: Angular Speed, Angular Acceleration Today: Reiew, Centripetal Acceleration, Newtonian Graitation i HW #6 due Tuesday, Oct 19, 11:59 p.m. Exam # Thursday, Oct 1 On material through (including) today s Lecture Tuesday s (Oct 19) material ilwill not be on Exam # 1

2 Rotational Motion Under Constant Angular Acceleration Just like for linear motion under constant acceleration, we hae kinematic equations for rotational motion. They are analogous Rotational Motion with Constant α Linear Motion with Constant a i t i at θ t i 1 t x x t i 1 at i θ i ax θ x a

3 Example: Extra An electric motor rotating a wheel at a rate of 100 re/min is turned off. Assume the wheel has a constant angular acceleration of.00 rad/s. (a) How long does it take for the wheel to come to a stop? (b) Through how many radians has the wheel turned during the time interal found in (a)? Extra: Suppose the wheel has a radius of 0.0 m. At t =.0 s, what tis the magnitude of the tangential ti speed and acceleration at a point on the edge of the wheel? 3

4 Centripetal Acceleration Consider a car driing in a circle at a constant speed of 5 mph. The speed is constant, t but what htabout tthe elocity? Recall: Velocity is a VECTOR. The car s speed is constant, but the elocity ector is constantly changing direction as it moes around the circle! If the elocity is changing, there must be an acceleration. 4

5 Centripetal Acceleration Initial i Δs Final f f i i Δθ r Δθ r f Note the direction of Δ (inwards ) Note: For constant speed, i and f Their magnitudes are the same, i = f =. Recall: f i a a a t t c r f i i differ only in their direction. r 5

6 Centripetal s. Tangential Acceleration Magnitude: a c r r a c At all points along the circle, a c points radially inwards For circular motion, centripetal acceleration is always present, een if the speed is not changing. It is the acceleration associated with the constant change in direction. Recall: If the speed is changing, there is a tangential acceleration. t a t t a t t Magnitude: a t r 6

7 Circular Motion: Total Acceleration a c + a t a t Centripetal Tangential Always exists it for For circular motion, only circular motion exists if speed is changing Magnitude of Total Acceleration for Circular Motion a a c at Direction of Total Acceleration Vector for Circular Motion Must do ector addition of a c and a t 7

8 Angular Quantities are VECTORS Technically, the angular speed of rotation is a ector: ω. The direction is gien by the Right Hand Rule. Point your thumb so that your fingers wrap around in the direction of motion. Your thumb then points in the direction of ω. 8

9 What Causes Centripetal Acceleration? Well, we hae Newton s Second Law of Motion: F ma If an object is moing in a circle, it is undergoing centripetal acceleration, and so this must hae been caused by a force! This force is called the centripetal force. [ NOT centrifugal force. Take PHY 404 to learn about that ] F c ma c m r Magnitude [SI: Newtons] Direction: F c The centripetal force is always directed radially inwards, just like the centripetal acceleration! It is perpendicular to the elocity. What happens if the centripetal force disappears? 9

10 Example: Car in a Circular Turn A car traels at a constant speed of 13.4 m/s (30 mph) on a leel circular turn of radius m. What minimum coefficient of static friction, μ s, between the tires and roadway will allow the car to make the turn without sliding? 10

11 Example: Roller Coaster! A roller coaster moes in a circular loop of radius R. (a) What speed must the coaster hae so that it will just make it oer the top of the track without any assistance from the track? (b) What speed will the coaster hae at the bottom? (c) What will be the normal force on a passenger at the bottom of the loop if R = 10.0 m? (This is your perceied weight.) 11

12 Newtonian Graitation Q: Is the same force responsible? A: Yes! Newton s Law of Uniersal Graitation If two particles with masses m 1 and m are separated by a distance r, a graitational force F acts along a line joining them, with magnitude G = kg 1 m 3 s 1 [ graitational constant ] F G m 1m [ graitational constant ] r Force is always attractie 1

13 Graitational Force m 1 m F 1 F 1 attractie graitational force 1 F 1 = F 1 The Third Law. Equal but opposite directions. Action/Reaction pair. 3 Magnitudes are equal, calculated with Newton s Law of Uniersal Graitation. Gauss s Law: The graitational force exerted by a uniform sphere on a mass located outside the sphere is the same as if the entire mass of the sphere were concentrated at its center. 13

14 Graitational Force on the Earth s Surface What is the graitational force on a mass m on the surface of the Earth? m M E R E R E = m M 598 E = kg By Gauss s Law: What is the acceleration? m E R E RE mm M ma G a G R E E F M E mm E G R E Plug in the numbers a = 980m/s 9.80 = g!! 14

15 What if Aboe the Surface of the Earth? 1000 km m Example: A mass m is R E = m R 1000 km aboe the E surface of the Earth. M E = kg M E What is its acceleration? By Gauss s Law: m F ma G mm ( R 10 E 6 m) R E km a = 733m/s 7.33 M < g!! E R E F = mg only applicable near the surface of the Earth!! 15

16 Next Class : Graitational Potential Energy (re isited), Kepler s Laws of PlanetaryMotion 16