PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 10 Last Lecture m1v1i + m 2v 2i = m1v1 f + m2v 2 f Elastic Collisions: v1i v 2i = (v1 f v 2 f ) Multi-part Collision Problems (conserve E or p) s = r Angular motion

( in radians) Angular Speed f i = = t t f ti (in rad/s) Can also be given in Revolutions/s Degrees/s Linear (tangential) Speed at r

s r vt = = t t v t = r ( in rad/s) Example 7.2 A race car engine can turn at a maximum rate of 12,000 rpm. (revolutions per minute). a) What is the angular velocity in radians per second.

b) If helicopter blades were attached to the crankshaft while it turns with this angular velocity, what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound. 1256 rad/s DATA: The speed of sound is 343a)m/s b) 27 cm Angular Acceleration Denoted by f i = t in rad/s rad/s Every point on rigid object has same

and Rotational/Linear Correspondence: x 0 v0 f vf a tt Rotational Motion Linear Motion ( 0 + f ) = t 2

( v0 + v f ) x = t 2 f = 0 + t v f =v0 + t 1 2 = 0 t + t 2 1 2 = f t t 2 1 2

x = v0 t + at 2 1 2 x = v f t at 2 2 f 2 = 02 2 +

2 vf v02 = + x 2 2 Constant a Constant Rotational/Linear Correspondence, contd Example 7.3 A pottery wheel is accelerated uniformly from rest to a rotation speed of 10 rpm in

30 seconds. a.) What was the angular acceleration? (in rad/s2) b.) How many revolutions did the wheel 0.0349 rad/s2 undergo during that a) time? b) 2.50 revolutions Linear movement of a rotating point Distance s = r Speed v t = r Acceleration

at = r Angles must be in radians! Different points have different linear speeds! Special Case - Rolling Wheel (radius r) rolls without slipping Angular motion of wheel gives linear motion of car

x =r Distance Speedv =r Acceleration a =r Example 7.4 A coin of radius 1.5 cm is initially rolling with a rotational speed of 3.0 radians per second, and comes to a rest after experiencing a slowing down of = 0.05 rad/ s2. a.) Over what angle (in radians) did the coin rotat b.) What linear distance did the coin move? a) 90 rad b) 135 cm

Centripetal Acceleration Moving in circle at constant SPEED does not mean constant VELOCITY Centripetal acceleration results from CHANGING DIRECTION of the velocity r r v a= t Acceleration points toward center of circle Derivation: acent = 2r = v2/r Similar

triangles: v s = v r v v s aavg = = t r t s arc length = r Small times: a=v = v

t v = r Using =v /r or 2 v acent = 2 r = r Forces Cause Centripetal Acceleration

Newtons Second Law r r F =m Radial acceleration requires radial force Examples of forces Spinning ball on a string Gravity Electric forces, e.g. atoms Example 7.5a An astronaut is in circular orbit around the

Earth. Which vector might describe the a) Vector A astronauts b) Vector B velocity? c) Vector C A B C Example 7.5b An astronaut is in circular

orbit around the Earth. Which vector might describe the a) Vector A astronauts b) Vector B acceleration? c) Vector C A B C Example 7.5c An astronaut is

in circular orbit around the Earth. Which vector might describe the gravitational force a) acting Vector on A the b) astronaut? Vector B c) Vector C A

B C Example 7.6a Dale Earnhart drives 150 mph around a circular track at constant speed. Neglecting air resistance, which vector best describes the frictional force exerted on the tires from contact with the

a) Vector A pavement? b) Vector B c) Vector C B A C Example 7.6b Dale Earnhart drives 150 mph around a circular track at constant

speed. Which vector best describes the frictional force Dale Earnhart experiences from the seat? a) Vector A b) Vector B c) Vector C B A C

Ball-on-String Demo Example 7.7 A puck (m=.25 kg), sliding on a frictionless table, is attached to a string of length 0.5 m. The other end of the string is fixed to a point on the table and the puck is sent revolving around the fixed point. It take 2 seconds to make a complete revolution. a) What is the acceleration of the puck? b) What is the tension in the string? a) 4.93 m/s2 b) 1.23 N DEMO: FLYING POKER CHIPS Example 7.8

A race car speeds around a circular track. a) If the coefficient of friction with the tires is 1.1, what is the maximum centripetal acceleration (in gs) that the race car can experience? b) What is the minimum circumference of the track that would permit the race car to travel at 300 km/hr? a) 1.1 gs b) 4.04 km (in real life curves are banked Example 7.9 A curve with a radius of curvature of 0.5 km on a highway is banked at an angle of 20. If the highway were

frictionless, at what speed could a car drive without sliding off the road? 42.3 m/s = 94.5 mph Example 7.11a Which vector represents acceleration? a) A b) E c) F d) B

e) I Example 7.11b If car moves at "design" speed, which vector represents the force acting on car from contact with road a) D b) E c) G d) I e) J

Example 7.11c If car moves slower than "design" speed, which vector represents frictional force acting on car from contact with road (neglect air resistance) a) B b) C c) E d) F e) I