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Physics Help >> Index to Physics Homework Help » Circular Motion
Circular motion is caused when an object experiences just the right amount of net force directed toward the center of the circle or curve. The acceleration of an object in uniform circular motion is always toward the center of the circle. The acceleration and the force causing it are called centripetal which means center-seeking. Centripetal force can be provided to a planet by gravity, to a car by friction of the wheels on the road, to a ball the the tension on a string. Since the direction of the velocity of the object is continuously changing toward the center of the circle, the object is said to experience centripetal acceleration, even if its speed is constant. (Searching for F = mv2 R ? See below.)
A. Speed and Acceleration
2. What is the rotational speed in m/s and in miles per hour of a person standing at the Earth's equator (R = 6.38 x 106 m)?
3. Tampa, Florida is at latitude of about 28º north. What is the distance traveled by a building in Tampa in one day in space as a result of the Earths rotation?
4. If the rotation of a planet with a radius of 6.04*106 m and it's free fall acceleration of 9.9 m/s2 increased to the the point that the centripetal acceleration was equal to the gravitational acceleration at the equator, what would be the tangential speed of a person standing at the equator?
5. Show that a point on the surface of the earth at latitude ø from the equator has an acceleration of magnitude 3.37*cosø cm/s2 relative to a reference frame not rotating with the earth.
6.
B. Horizontal Circular Motion Problems
2. A rock is whirled on the end of a string in a horizontal circle of radius R and period T. If the radius is halved while keeping the period constant, what happens to the centripetal acceleration of the rock?
3. A boy is whirling a yo yo on a string in a horizontal circle. What happens to the tension on the string when he whirls it twice as fast?
4. A mass m = 2.8 kg on a frictionless table is attached to a hanging mass M = 7.9 kg by a cord through a hole in the table. Find the speed with which m must move (in a circle of radius r = 0.19 m) in order for M to stay at rest?
5. A flea stands on the end of a 1.0 cm long sweep second hand of a clock that rests horizontally on a table. What is the minimum coefficient of static friction which would allow the flea to stay there without slipping?
6. What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is 25 km/h and the coefficient of static friction between the tires and the road is 0.28?
7. A rock of mass 4.0*102 g is tied to one end of a string that is 2.0 m in length. Holding the other end above his head, a boy swings the rock around in a circle whose plane is parallel to the ground. a. If the string can withstand a maximum tension of 4.5 N before breaking, what angle to the vertical does the string reach just before breaking? b. At what speed is the rock traveling just as the string breaks?
8. What is the maximum speed with which a 1050 kg car can round a turn of the radius 70 m on a flat road if the coefficient of friction between tires and road is 0.80? Is this result independent of the mass of the car?
9. On an ice rink two skaters of equal mass grab hands and spin in a mutual circle once every four seconds. If we assume their arms are each 0.76 m long, how hard are they pulling on one another, assuming their individual masses are 55.0 kg?
10. An amusement park ride consists of a large cylinder on radius R spinning about its vertically oriented axis of symmetry. The rider is held to the inner cylinder wall by static friction as the bottom of the cylinder is lowered. Friction at the interface between the cylinder and the rider is characterized by the coefficient of μs. What conditions must be placed on the period of rotation to ensure the rider does not slip down the wall when the bottom is removed?
11. A ball rolls around a horizontal circle at height y inside a frictionless hemisphere bowl of radius R. Calculate an expression for the ball’s velocity in terms of R, y and g (in order for the ball to maintain this horizontal circular path).
12. A train traveling at a constant speed rounds a curve of radius 225 m. A chandelier suspended from the ceiling swings out to an angle of 20.0º throughout the turn. What is the speed of the train?
C. Vertical Circular Motion Problems
2. A jet fighter pilot knows he is able to withstand an acceleration of 9g before blacking out. The pilot points his plane vertically down while traveling at Mach 3 speed and intends to pull up in a circular maneuver before crashing to the ground. a) Where does the maximum acceleration occur in the maneuver? b) What is the minimum radius the pilot can take?
3. A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to 3.0 times her weight as she goes through the dip. If r = 25.0 m, how fast is the roller coaster traveling at the bottom of the dip?
4. Driving in your car with a constant speed of 12 m/s, you encounter a bump in the road that has a circular cross-section. If the radius of curvature of the bump is 35 m, find the apparent weight of a 75-kg person in your car as you pass over the top of the bump.
5. What is the minimum speed of a roller coaster at the top of a 39.0 m vertical loop if the passengers are "weightless" at that point.
6. A ball of a mass 4.0 kg is attached to the end of a 1.2 cm long string and whirled around in a circle that describes a vertical plane. a. What is the minimum speed that the ball can be moving at and still maintain a circular path? Provide a free body diagram. b. At this speed, what is the maximum tension in the string? Provide a free body diagram. c. If the ball was swung in a horizontal circle at this speed, what angle would the string make with the vertical?
7. How do you find the tension in the string of a ball traveling in a vertical circle at the 45 degree angle?
8. A bridge is in the shape of an arch having the radius of curvature of 41. m. What is the maximum speed that a car can travel across the bridge without 'getting some air'?
D. Banked Curves
2. If a curve with a radius of 65 m is properly banked for a car traveling 75 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 90 km/h?
3. A Car is driven around a circle with a radius of 200m, bank angle 10 degrees. The static frictional coefficient is 0.60. Calculate the maximum velocity the car can travel (Vmax).
4. An airplane is flying in a horizontal circle at a speed of 460 km/h. If its wings are tilted 40° to the horizontal, what is the radius of the circle in which the plane is flying? Assume that the required force is provided entirely by an "aerodynamic lift" that is perpendicular to the wing surface.
E. Particles in Fields
1. Photons of wavelength 450 nm are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius 20 cm by a magnetic field whose strengths is 2*10-5 T. What is the work function of the metal?
Selected solutions are printed below. For solutions to all the problems on this page click here.
A1. ac = (3.0 m/s)2/(0.5 m) ac = 18 m/s2 b. Fc = mac = 36 J
For solutions to all the problems on this page click here.
B10. Frictional force, Ff, must be equal to or larger than the weight of the rider:
Ff ≥ mg where m is the mass of the rider.
Since Ff = μsN where N is normal force of the cylinder wall on the rider
μsN ≥ mg
Normal force is the centripetal force the cylinder applies to the rider:
N =m4π2R/T2 where T is the period of rotation.
Substituting,
μsm4π2R/T2 ≥ mg
Simplifying and rearranging,
μs4π2R/g ≥ T2
T ≤ 2π√(μsR/g)
For solutions to all the problems on this page click here.
C8.
The net force on the passenger is W - N, where W is weight and N is normal force.
The car will leave the surface when the car exceeds a speed where normal force is zero.
The net force is centripetal force, so
mv2/r = W - N
mv2/r = mg - 0
Rearranging and simplifying,
v = √(rg) = √(41*9.81) = 20.1 m/s
The maximum speed is 20.1 m/s.
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A space station is moving at constant velocity through deep space without being influenced by gravitational fields. It spins around its hub as it travels. An astronaut standing at rest relative to the space station would occupy which of the indicated positions inside the space station? 