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AL Physics/Circular Motion/P.1
PLK Vicwood K.T. Chong Sixth Form College
Form Six AL Physics Exercise
Uniform Circular Motion
1.
A rotating fan completes 1200 revolutions every minute. Consider a point on the tip of the blade,
which has a radius of 0.15 m. (a) Through what distance does the point move in one revolution?
(b) What is the speed of the point? (c) What is its acceleration? [(a)94 cm (b)19 m/s (c)2400
m/s2]
2.
In the Bohr model of the hydrogen atom, the electron revolves in a circular orbit around the
nucleus. If the radius is 5.3 × 10-11 m and the electron makes 6.6 × 1015 rev/s, find (a) the speed
of the electron, (b) the acceleration of the electron, and (c) the force acting on the electron. (This
force is the result of the attraction between the positively charged nucleus and the negatively
charged electron. Mass of electron = 9.1 × 10-31 kg) [(a) 2.2 × 106 m/s (b) 9.11 × 1022 m/s2
(c) 8.29 × 10-8 N]
3.
An object of mass 4 kg moves round a circle of radius 6 m with a constant speed of 12 m/s.
Calculate (i) the angular speed, (ii) the force towards the centre. [(i) 2 rad/s (ii) 96 N]
4.
Calculate the force necessary to keep a mass of 0.2 kg moving in a horizontal circle of radius 0.5
m with a period of 0.5 s. What is the direction of the force? [15.8 N, toward the center of the
circle]
5.
A 10.7-kN car traveling at 13.4 m/s attempts to round an unbanked curve with a radius of 61.0
m. (a) What force of friction is required to keep the car on its circular path? (b) What minimum
coefficient of static friction between the tires and road is required. [(a) 3200 N (b) 0.30]
6.
A small object is placed 13.0 cm from the center of a phonograph turntable. It is observed to
remain on the table when it rotates at 33 1/3 rev/min but slides off when it rotates at 45.0 rev/min.
Between what limits must the coefficient of static friction between the object and the surface of the
turntable lie? [0.29]
7.
A circular curve of highway is designed for traffic moving at 60 km/h. (a) If the radius of the curve
is 150 m, what is the correct angle of banking of the road? (b) If the curve were not banked,
what would be the minimum coefficient of friction between tires and road that would keep traffic
from skidding at this speed? [(a) 10.7º; (b) 0.19]
8.
An airplane is flying in a horizontal circle at a speed of 482 km/h. The wings of the plane are tilted
at 38.2° to the horizontal; see Figure 8. Find the radius of the circle in which the plane is flying.
Assume that the centripetal force is provided entirely by the lift force perpendicular to the wing
surface. [2.32 km]
9.
A 1.34-kg ball is attached to a rigid vertical rod by means of two massless strings each 1.70 m
long. The strings are attached to the rod at points 1.70 m apart. The system is rotating about the
axis of the rod, both strings being taut and forming an equilateral triangle with the rod, as shown in
Figure 9. The tension in the upper string is 35.0 N. (a) Find the tension in the lower string. (b)
Calculate the net force on the ball at the instant shown in the figure. (c) What is the speed of the
ball? [(a) 8.7 N; (b) 37.9 N; (c) 6.45 m/s]
AL Physics/Circular Motion/P.2
10.
A small block of mass m slides along the frictionless loop-the-loop track shown in Figure 10. (a)
The block is released from rest at point P. What is the net force (in terms of m and g) acting on it
at point Q? (b) At what height above the bottom of the loop should the block be released so that
it is on the verge of losing contact with the track at the top of the loop? [(a) 8.06 mg, at 82.9° left
of vertical. (b) 5R/2]
11. A car moves at a constant speed on a straight but hilly road. One section has a crest and dip of
the same 250-m radius; see Figure 11. (a) As the car passes over the crest, the normal force on
the car is one-half the 16-kN weight of the car. What will be the normal force on the car as it
passes through the bottom of the dip? (b) What is the greatest speed at which the car can move
without leaving the road at the top of the hill? (c) Moving at the speed found in (b), what will be
the normal force on the car as it moves through the bottom of the dip? [(a) 24 kN; (b) 49.5 m/s;
(c) 32 kN]
12. A conical pendulum is formed by attaching a 53-g pebble to a 1.4-m string. The pebble swings
around in a circle of radius 25 cm. (a) What is the speed of the pebble? (b) What is its
acceleration? (c) What is the tension in the string? [(a) 0.67 m/s; (b) 1.78 m/s2; (c) 0.53 N]
13. A 666.8-N student on a steadily rotating Ferris wheel (Figure 13) has an apparent weight of
555.7 N at the highest point. (a) What is the student's apparent weight at the lowest point? (b)
What would be the student's apparent weight at the highest point if the speed of the Ferris wheel
were doubled? [(a) 777.9 N (b) 222.3 N]
14. A certain string can withstand a maximum tension of 40.9 N without breaking. A child ties a 3.6N stone to one end and, holding the other end, whirls the stone in a vertical circle of radius 0.88
m, slowly increasing the speed until the string breaks. (a) Where is the stone on its path when the
string breaks? (b) What is the speed of the stone as the string breaks? [(a) At the bottom of the
circle (b) 9.4 m/s]
15. A bicyclist travels in a circle of radius 25 m at a constant speed of 8.7 m/s. The combined mass of
the bicycle and rider is 85 kg. Calculate the force - magnitude and angle with the vertical - exerted
by the road on the bicycle. [870 N; 17°]
16. A model airplane of mass 0.75 kg is flying at constant speed in a horizontal circle at one end of a
33-m cord and at a height of 18 m. (Figure 16) The other end of the cord is tethered to the
ground. The airplane makes 4.4 rev/min and the lift is perpendicular to the unbanked wings. (a)
What is the acceleration of the plane? (b) What is the tension in the cord? (c) What is the lift
produced by the plane’s wings? [(a) 5.87 m/s2 (b) 5.24 N (c) 10.2 N]
17. The particle m in Figure 17 is moving in a vertical circle of radius r inside a track. There is no
friction. When m is at its lowest position, its speed is v 0. (a) What is the minimum value v m of v 0
for which m will go completely around the circle without losing contact with the track? (b) Suppose
v 0 is 0.775v m. The particle will move up the track to some point at P at which it will lose contact
with the track and travel along a path shown roughly by the dashed line. Find the angular position θ
of point P. [(a) 5gr (b) 19.5º]
AL Physics/Circular Motion/P.3
18. A racing car of mass 500 kg starts from rest and accelerates at 6.0 m/s2 along a straight horizontal
road for a distance of 150 m. It then enters at a constant speed a horizontal circular curve of radius
200 m.
(i) What is its speed through the curve? (ii) What is the magnitude and direction of the resultant
horizontal force acting on the racing car while it is rounding the curve? (iii) If, while on the curve, the
racing car accelerates forwards at 3.0 m/s2, what is the resultant horizontal force acting on the car at
the time it begins to accelerate forwards, and in which direction does it act? Illustrate your answer
with a diagram. (iv) State two parameters that limit the safe speed at which the racing car can travel
around a horizontal curve of a given radius. (v) Show that by suitable banking the road can be
made perfectly safe for racing cars cornering at a particular speed. Calculate the banking angle
needed for the speed at which the racing car entered the curve of radius 200 m. [(i) 42 m/s (ii)
4500 N (iii) 4743 N, 18º (v) 43º]
19. In a demonstration, a cord is tied to a bucket which contains 1500 cm3 of water and the bucket is
swung in a vertical circle of radius 1 m. The mass of the bucket is 0.25 kg. (Water density = 1000
kg/m3)
(a) What must be the minimum velocity of the bucket at the ‘lowest’ point of the circle if no water is
to spill from the bucket at the top of the circle?
(b) Assume that the velocity of the bucket at the highest point of the circle is 5 m/s.
(i) Draw a free body diagram for
(I) the water;
(II) the bucket
at the highest point of the circle.
(ii) At the highest point of the circle, what is the origin(s) of the centripetal force acting on
(I) the water;
(II) the bucket.
(iii) Find, at the highest point of the circle,
(I) the reaction force between the bottom of the bucket and the water;
(II) the tension in the cord.
[(a) 7.07 m/s (b) (iii) (I) 22.5 N (II) 26.25 N]
AL Physics/Circular Motion/P.4
Figure 8
60º
1.70 m
35.0 N
1.34 kg
38.2º
Figure 9
P
Figure 10
250 m
R
5R
Q
250 m
Figure 11
Figure 13
model
airplane
Figure 16
P
θ
r
v0
Figure 17
m