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 centerseeking. Centripetal force can be provided to a planet by gravity, to a car by friction between wheels and the road, or 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 though its speed is constant. (Searching for F = mv2 R ? See below.)
A. Speed and Acceleration
(For answers see below.)
1. A 2.0 kg mass swinging at the end of a 0.50 m string is traveling
3.0 m/s. What is the
a. centripetal acceleration of the mass?
b. centripetal force on the mass?
2. A person standing at the Earth's equator has what rotational speed? (R = 6.38 x 10^{6} m)
3. A building is located 28º north of Earth's Equator. What distance does it travel as a result of the Earths rotation?
4. A planet has a radius of 6.04*10^{6} m and free fall acceleration of 9.9 m/s^{2}. What would be the tangential speed of a person standing at the equator, if its rotation increased to the point that the centripetal acceleration was equal to the gravitational acceleration?
5. If a point on the surface of the earth at latitude ø from the equator what is its acceleration relative to a stationary reference frame not rotating with the earth?
6. 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?
B. Horizontal Circular Motion Problems
(For answers see below.)
1. A stopper tied to the end of a string is swung in a horizontal
circle. If the mass of the stopper is 13.0 g, and the string is
93.0 cm, and the stopper revolves at a constant speed 10 times in 11.8
s,
a. what is the tension on the string?
b. what would happen to the tension on the string if the mass was
doubled and all other quantities stayed the same?
c. what would happen to the tension on the string if the period
was doubled and all other quantities stayed the same?
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 0.19 m cord passes through a hole in a table.. The cord attaches a mass m = 2.8 kg on the frictionless surface to a hanging mass M = 7.9 kg. Find the speed with which m must move in a circle in order for M to stay at rest?
5. A clock rests horizontally on a table with a pebble balanced at the end of its 1.0 cm long second hand.. What is the minimum coefficient of static friction which would allow the pebble to stay there without slipping?
6. What is the smallest radius of an unbanked (flat) track around which a motorcyclist 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*10^{2} g is tied to one end of a string that is 2.0 m in length and swung 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. A 1050 kg car travels around a turn of radius 70 m on a flat road. If the coefficient of friction between tires and road is 0.80 what is the maximum speed the car can travel without slipping? Does this result dependent on the mass of the car?
9. Two ice skaters of equal mass grab hands and spin in a circle once every four seconds. Their arms are 0.76 m long, and they each have a mass of 55.0 kg. How hard are they pulling on one another?
10. A hollow vertical cylinder with radius R spins about its vertical axis of symmetry. A stone is held to the inner cylinder wall by static friction. Express the period of rotation in terms of the radius and the coefficient of friction, μ_{s}.
11. A coin slides around a horizontal circle at height y inside a frictionless hemisphere bowl of radius R. Derive the coin’s velocity in terms of R, y and g.
12. A train travels at a constant speed around a curve of radius 225 m. A ceiling lamp at the end of a light cord swings out to an angle of 20.0º throughout the turn. What is the speed of the train?
C. Vertical Circular Motion Problems
(For answers see below.)
1. A ball with a mass of 130 g is swung at the end of a string
93.0 cm in length. The ball is whirled in a vertical circle at
4.00 revolutions per second.
a. What is the tension on the string at the bottom of the loop?
b. What is the tension on the string at the top of the loop?
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. What is the apparent weight of a 75kg person driving a car with a constant speed of 12 m/s over a bump with a circular crosssection and radius of curvature of of 35 m?
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 hill 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 hill
without 'getting some air'?
D. Banked Curves
(For answers see below.)
1. Determine the minimum angle at which a road should be banked so that a car traveling at 20.0 m/s can safely negotiate the curve if the radius of the curve is 200.0 m.
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.
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, and force is provided by lift that is perpendicular to the wing surface. What is the radius of the circle?
ANSWERS to CIRCULAR MOTION PROBLEMS
Selected solutions are printed below.
For solutions to all the problems on this page click here.
A1.
a. a_{c} = v^{2}/r
a_{c} = (3.0 m/s)^{2}/(0.5 m)
a_{c} = 18 m/s^{2}
b. F_{c} = ma_{c} = 36 J
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B10.
Frictional force, F_{f}, must be equal to or larger than the weight of the stone:
F_{f }≥ mg where m is the mass of the stone.
Since F_{f }= μ_{s}N where N is normal force of the cylinder wall on the stone
μ_{s}N ≥ mg
Normal force is the centripetal force the cylinder applies to the stone:
N =m4π^{2}R/T^{2} where T is the period of rotation.
Substituting,
μ_{s}m4π^{2}R/T^{2} ≥ mg
Simplifying and rearranging,
μ_{s}4π^{2}R/g ≥ T^{2}
T ≤ 2π√(μ_{s}R/g)
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C8.
The net force on the car 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
mv^{2}/r = W  N
mv^{2}/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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