Ch 7 Circular motion, gravity and torque

1. Ch 7 Circular motion, gravity and torque

2. Tangential velocity &centripetal acceleration • useful to understand how  &  of a rotating object is related to linear speed and linear acceleration • objects in circular motion have a tangential speed (vt) • vt: the instantaneous linear speed of an object directed along the tangent to the objects circular path • two objects on a rigid rotating body have the same yet different vt at different radius

3. vt cont. • increase radius, increase vt • t is the same for both but increase radius ==> increase dist. travel per 1 rev. • vt=s/t (s=linear dist) =vt/r (r=radius) • vt = 2r/T

4. Board work • A golfer has a max. angular speed of 6.3rad/s for her swing. She choose between two drivers, one placing the club head 1.9m from her axis of rotating and the other placing it at 1.7m from the axis. • a. cal vt for each. b. which will hit the ball farther? • a. 12m/s & 11m/s. b. 1.9m

5. Tangential acceleration • tangent to the circular and is instantaneous • linear acceleration of an object directed along the tangent to the objects circular path • vt=r ( divide both by t) • vt/t=r/t (t small) • at = r ( units  = rad/s2)

6. Board work • a yo-yo has a tangential acceleration of .98m/s2 when it is released. the string is wound around a central shaft of radius .35cm. what is the  of yo-yo? • 2.8x102rad/s2 • a centrifuge starts from rest and accelerates to 10.4rad/s in 2.3s. what is the tangential acceleration of a vial that is 4.7 cm from the center? • .21m/s2

7. Centripetal acceleration(ac) • object moving in a circle at constant speed still has an acceleration due to its  in direction. • ac: acceleration directed towards the center of a circular path

8. ac equation • ac=v2/r = r2

9. Board work • a cylindrical space station with a 115m radius rotates around it’s longitudinal axis at an angular speed of .292rad/s. calculate the centripital acceleration on a person at the following location. a. center b. 1/2way c. at the rim • a. 0m/s2 b. 4.90m/s2 c. 9.80m/s2 • The cylindrical tub of a washing machine has radius of 34cm. During the spin cycle, the wall of the tube rotates with a tangential speed of 5.5 m/s. Calculate the centripetal acceleration of the clothes sitting against the tub. • 89 m/s2

10. at & ac • are perpendicular • at: is due to changing speed • ac: due to the change of direction • find the total ( at & ac) use the Pythagorean theorem: atotal = at2 + ac2 =[ac/at]tan-1

11. Centripetal force (Fc) • force that maintains circular motion • ball moving in a circle: v due to  in direction • ac is inward: ac=vt2/r • Fc is used to  an objects straight line inertia • Fc=mac =mr2=mv2/r

12. Ff & Fc • car around race track: Ff is the force that supplies the Fc necessary to travel in a circle • Fc=Ff • mv2/r=mg ( cancel m and rearrange to get v isolated) • v = gr

13. Board work • an astronaut who weighs 735N on earth is at the rim of a cylindrical space station with a 73m radius. the space station is rotating at an angular speed of 3.5 rpm. evaluate the force that maintains the circular motion of the astronaut. • 732N • the moon ( mmoon=7.36x1022kg) orbits earth at a range of 3.84x105km with a period of approximately 28 days. determine the force that maintains the circular motion of the moon. • 1.90x1020N

14. Fc cont. • force directed towards the center is necessary for circular motion • if the force is lost the object leaves at an tangent to the circular motion

15. Newton’s law of gravitation G=6.67x10-11 N-m2/kg2 • gravitational force is inversely proportional to the distance2 between two masses • gravitational force (Fg) is localized to the center of a spherical mass • mg = mi gravitational mass = inertial mass

16. Cavendish exp. • Cavendish used a torsion balance to cal G in the equation Fg=Gm1m2/d2 • G = 6.673 x 10-11 Nm2/kg2 • the relationship between Fg and distance is an inverse square relationship

17. Board work • cal g for the earth. ( Me=6.0x1024kg, Re=6.4x106m • 9.77m/s2 • find the gravitational force exerted on the moon(mmoon=7.36x1022kg) by the earth when the distance between centers is 3.84x108m • 1.99 x 1020N

18. Motion in space • Kepler’s laws: developed model of how planets moved around the sun • Geocentric: earth centered, Ptolemy and epicycles. • Helocentric- sun centered, Copernicus and Galelio. • Tycho Brahe made very accurate observations of Mars orbit around the sun. • Kelper used Brahe’s data to develop his laws of planetary motion (helocentric).

19. Kepler’s laws of planetary motion • First law: Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points. • Second law: An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals • Third law: The square of a planet’s orbital period (T2) is proportional to the cube of the average distance (r3) between the planet and the sun. T2 prop. r3

20. Development of Kepler’s laws • Kepler used the motion of Mars to determine planets moved in ellipses. • As a result the law of equal areas predicted planets closer to the sun moved faster than father away. • Third law (law of harmony) relates the orbit period and distances of one planet to all the other. • Can be used to predict planet location • Determine satellite location for specific orbits • T12/T22 = r13/r23

21. Kepler’s and Newtons laws • Newton used Keplers laws to support his law of gravitation. • Fc = Fg, Gmm/r2 = mvt2/r • vt = 2r/T ( r = center to center of objects) • T2 =(42/Gm)r3 m = mass of central object. Mass of planet or satellite that is in orbit does not affect speed or period. • Planetary data pg 250 table 1 • A large planet orbiting a distant star is discovered. The planet’s orbit is nearly circular and close to the star. The planets orbital distance is 7.50x1010m, its period is 105.5 days. Calculate the mass of the star. • 3.00x1030kg

22. Weight and weightlessness • Weight: measure your downward force • Newton’s third law action reaction. • When in up-ward motion your measured weight changes. • Accelerating up-ward reaction force causes your measure force to increase. • Accelerating down-ward reaction force causes your measured force to decrease. • “Weightlessness”: when in free fall. No normal force acting on you.

23. Weightlessness in space • Force of gravity keeps objects in orbit, space ships are accelerating at the same rate as it passengers so there is sense of weightlessness like free fall on earth. • Gravity is used by our bodies to help push blood and maintain muscle tone and bone density. • Without it can cause health risks.

24. Rotational motion • Uniform circular motion: objects move in circular paths at constant speed. • Objects velocity (direction) is changing constantly. Ac and Fc directed inward. • Explore how to measure the ability of a force to rotate an object. Rotation motion is the entire object, circular motion is a point on a rotating object.

25. Torque () • A quantity that measures the ability of a force to rotate an object around the axis of rotation. • Depends on the force and lever arm. • Lever arm: perpendicular distance from the axis of rotation to a line along the direction of the force. • As distance decreases the force to apply the same torque increases. • Lever arm depends on the angle the force is applied relative to the perpendicular.

26. Torque () • Torque = force x lever arm,  = Fd sin  • Units Nm • Torque is a vector quantity. (+ or - due to direction of rotation) • + for ccw rotation and - for cw rotation. • net =  = 1 + 2 + …. • A student pushes with a minimum force of 50.0N on the middle of a door to open it. What minimum force must be applied at the edge of the door in order to open it? • 25.0N

27. Simple machines • Device that transmits or modifies force, usually by changing the force applied (direction or magnitude). • 6 simple machines; lever, pulley, incline plane, wheel and axil, wedge and screw. (pg 259 table 2) • Mechanical advantage: compare how large the output force is relative to the input force. • MA = output force = Fout = din • input force Fin dout • The longer the input lever arm as compared with the output lever arm, the greater the MA

28. Machines alter force and distance • Work is still constant. • Efficiency measure how well a machine works. Frictions dissipates energy and decreases efficiency. • Eff = Wout/Win • Less than 1.