Understanding Uniform Circular Motion (U.C.M.): Exploring Centripetal Acceleration and Forces

1. Uniform Circular MotionPart 2 aka…U.C.M.: An object moving in a perfect circle at a constant speed.

2. The greater the speed, the faster the object changes direction, the greater the centripetal acceleration. • If comparing two spots on a single rotating object, when placed at a spot of greater radius, the speed is greater, the centripetal acceleration is greater; therefore, the Net force is greater. • Period is the same for both people!

3. What’s the centripetal acceleration of the bobsled during each section? Speed doesn’t change… assume very little friction. • ac = v2/r • ac = 35 m/s2 for 33 m radius • ac= 48 m/s2 for 24 m radius • As the radius decreases (gets smaller/tighter), the ac increases.

4. The model airplane of mass 0.90 kg moves at a constant speed on a circle path that is parallel to the ground. The guideline is a length of 17 m. Determine tension FT in the guideline for speeds 19 m/s and 38 m/s. • FT = 19 N • FT = 76 N • When v is doubled, Fc is… • Fc = mac = mv2/r • ?Fc= m(2v)2/r • Fc is 22 or 4x greater!

5. When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? • Friction • Tension • Normal force • Air resistance • Gravity Slide 6-8

6. Answer: • When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? • Friction • Tension • Normal force • Air resistance • Gravity Slide 6-9

7. A car approaches a level, circular curve with a radius of 45.0 m. If the concrete is dry (µs=0.900), what is the maximum speed at which the car can “negotiate” the curve at a constant speed?

8. Did you get…??? • ∑F = ma • ∑F = Ff = Fc • µsFN = Fc • µsFg = Fc • µsmg = Fc • µsmg = mv2/r (aha… mass cancels) • µsg = v2/r • v2= µsgr • v=(µsgr) • v=[(0.9)(9.80m/s2)(45m)] • v = 19.9 m/s

9. Example Problem A level curve on a mountain road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40?

10. Did you get…??? • ∑F = ma • ∑F = Ff = Fc • µsFN = Fc • µsFg = Fc • µsmg = Fc • µsmg = mv2/r (again… m cancels) • µsg = v2/r • v2= µsgr • v=(µsgr) • v=[(0.4)(9.80m/s2)(150m)] • v = 24.2 m/s

11. Example Problem In the hammer throw, an athlete spins a heavy mass in a circle at the end of a chain, then lets go of the chain. For male athletes, the “hammer” is a mass of 7.3 kg at the end of a 1.2 m chain. A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the chain, what is the tension in the chain?

12. Hammer Throw… and did you get? • m = 7.3 kg, r = 1.2 m, v = 29 m/s • FT = ? • FT = m∙v2/r = (7.3 kg) ∙ (29 m/s)2 / (1.2 m) • FT = 5116 N • {or about 5100 N}

13. Example Problem: Driving Over A Rise • A car of mass 1500 kg goes over a hill at a speed of 20 m/s. The shape of the hill is approximately circular, with a radius of 60 m, as in the figure to the right. When the car is at the highest point of the hill, • What is the force of gravity on the car? • What is the magnitude and direction of the normal force of the road on the car at this point?

15. Apparent weight = the magnitude of the force that supports you (FN). At top: FN + Fg = Fnet = Fc= mv2/r so, FN = mv2/r- Fg(feel lighter)At bottom: FN - Fg = Fnet = Fc = mv2/rso, FN = Fg + mv2/r(feel heavier)

16. Example Problem: Loop-the-Loop A roller coaster car goes through a vertical loop at a constant speed. For positions A to E, rank order the: • centripetal acceleration • normal force • apparent weight

17. Answer

18. The Globe of Death is a spherical cage in which motorcyclists ride in circular paths at high speeds. One outfit claims that riders achieve a speed of 60mph in a 16-ft diameter sphere. 1000 m = 0.621 miles, 1 inch = 2.54 cm, 100 cm = 1m • What would be the period for this motion? • What would be the apparent weight of a 60.0 kg rider at the bottom of the sphere?

19. Answer