Problems on uniform circular motion
Problem 1:
A 2 kg ball on a string is rotated about a circle of radius 10 m. The maximum tension
allowed in the string is 50 N. What is the maximum speed of the ball?
Problem 2:
During the course of a turn, an automobile doubles its speed. How much additional
frictional force must the tires provide if the car safely makes around the curve?
Problem 3:
A satellite is said to be in geosynchronous orbit if it rotates around the earth once every
day. For the earth, all satellites in geosynchronous orbit must rotate at a distance of
4.23×107 meters from the earth's center. What is the magnitude of the acceleration felt
by a geosynchronous satellite?
Problem 4:
The maximum lift provided by a 500 kg airplane is 10000 N. If the plane travels at 100
m/s, what is its shortest possible turning radius?
Problem 5:
A popular daredevil trick is to complete a vertical loop on a motorcycle. This trick is
dangerous, however, because if the motorcycle does not travel with enough speed, the
rider falls off the track before reaching the top of the loop. What is the minimum speed
necessary for a rider to successfully go around a vertical loop of 10 meters?
Uniform circular motion
A. Speed and Acceleration
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. What is the angular speed in rad/sec and tangential speed in m/s of a person
standing at the Earth's equator (R = 6.38 x 106 m)?
B. Horizontal Circular Motion Problems
1. A stopper tied to the end of a string is swung in a horizontal circle. If the mass of
the stopper is 0.13 kg, and the string is 0.93 m, 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 yoyo 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 0.2 m 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 6 m/s and the coefficient of static friction between the tires and the
road is 0.28?
7. A rock of mass 4.0 x 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. If the string can withstand a maximum tension of 4.5 N
before breaking. 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 linear speed to ensure the rider does not slip down the wall when the
bottom is removed?
C. Vertical Circular Motion Problems
1. A ball with a mass of 0.130 kg is swung at the end of a string 0.93 m 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 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?
3. 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.
D. Banked Curves
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 20 m/s, what
must be the coefficient of static friction for a car not to skid when traveling at 30
m/s?
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 110 m/s. 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. Satellites in circular orbits, apparent weightlessness & artificial gravity
1. Suppose the velocity of the space station is 36 m/s. What must be the value of radius
‘r’ for the astronauts to weigh the same as on earth?
2. The moon orbits the earth at a distance of 3.85 x 108 m. Find the time period of the
moon’s motion around the earth in seconds and also express it in days. What is the
centripetal acceleration of the moon in orbit? The mass of the Earth is 5.98 x 1024 kg.
More Problems on Ch 5 = More Fun!!
1. The Earth orbits the Sun at an average distance of 1.5 x 1011 m, in
approximately 365.25 days (1 day = 86,400 sec. What is the Earth’s a) Angular
speed in rad/sec? b) Tangential speed ‘v’? and c) the mass of the Sun?
(G = 6.67 x 10-11N.m2/kg2)
2. A daredevil motorcyclist is going around a vertical circular loop of height 18 m.
What is the minimum speed he should have so he can safely make it without
falling? Does the mass matter?
3. An object of mass 0.4 kg is being swung in a circle of radius 0.5 m. If it makes 2
revolutions per second, what is the a) Tangential speed, b) Centripetal Force, c)
Tension in the string at the top, d) Tension at the bottom