UNIFORM CIRCULAR MOTION
 A light inextensible string of length L is fixed at its upper end and support a mass m at the other end. m is rotated at horizontal plane or radius r as shown. The maximum tension the string can withstand without breaking is 2N. Assuming the string breaks when the radius is maximum, calculate the velocity of the mass when the string breaks, given that L 1.25m, and m= 0.1kg.
 The diagram below shows a mass m, which is rotated in a vertical circle. The speed of the mass is gradually increased until the string breaks. The string breaks when the mass is at its lowest position A and at a speed of 30ms^{1}. Point a is 5m above the ground.
 a) Show on the diagram.
 i) The initial direction of the mass at the point the string breaks.
 ii) The path of the mass from A until it strikes the ground at a point b.
 b) Calculate;
 i) The time the mass takes to reach the ground after breaking off.
 ii) The horizontal distance the mass travels before it strikes the ground.
iii) The vertical velocity with which the mass strikes the ground.
 State the principle by which a speed governor limits the speed of a vehicle.
 The rear wheel of a certain car has a diameter of 40cm. At a certain speed of the car, the wheel makes 7 revolutions per second. A small stone embedded in the tyre tread flies off initially at an angle of 45^{0} to the ground. Determine the initial velocity of the pebble (take p = ^{22}/_{7})
 a) Explain why a pail of water can be swung in a vertical circle without the water pouring out.
 b) A car of mass 1,200kg is moving with a velocity of 25m/s around a flat bend of radius 150m. Determine the minimum frictional force between the tyres and the road that will prevent the car from sliding off.
 a) The fig shows the diagram of a set up to investigate the variation of centripetal force with the radius r of the circle in which a body rotates. Describe how the set up can be used to carry out the investigation
 b) The table shows results obtained from an investigation similar to the one in part (a)
Mass, m(g)  60  50  40  30  20 
Radius, r (cm)  50  41  33  24  16 
 i) Plot a graph of force, F (yaxis) on the body against the radius r (in metre)
 ii) Given that the mass of the body is 100g, use the graph to determine the angular velocity.
 A small object moving in a horizontal circle of radius 0.2m makes 8 revolutions per second. Determine its centripetal acceleration.
 (a) Define the term angular velocity.
(b) A body moving with uniform angular velocity found to have covered an angular distance 170 radians in t seconds. Thirteen seconds later it is found to have covered a total angular distance of 300 radians. Determine t
(c) Fig. 8 shows a body of mass m attached to the centre of rotating table with a string whose tension can be measure. (This device for measuring the tension is not shown in the figure).
The tension, T, on the string was measured for various values of angular velocity,
The distance r of the body from the centre was maintained at 30cm. Table 1 shows the results obtained.
Table 1
Angular Velocity (rads ^{1})  2.0  3.0  4.0  5.0  6.0 
Tension T (N)  0.04  0.34  0.76  1.30  1.96 
 i) Plot the graph of T (yaxis) against W^{2}
 ii) From the graph, determine the mass, m, of the body given that T= mw^{2}rC where C is a constant
iii) Determine the constant C and suggest what it represents in the set up.
 A child of mass 20kg sits on a swing of length 4m and swings through a vertical height of 0.9m as shown in the figure below.
Determine the:
 i) Speed of the child when passing through the lowest point.
 ii) Force exerted on the child by the seat of the swing when passing through the lowest point.
 a) State what is meant by centripetal acceleration?
 b) Figure 12 shows masses A, B and C placed at different points on a rotating table. The angular velocity, @ of the table can be varied.
 i) State two factors that determine whether a particular mass slides off the table or not.
 ii) It is found that masses slide off at angular velocities @_{A,} @ @_{c} of in decreasing order.
 c) A block of mass 200g is placed on a frictionless rotating table while fixed to the centre of the table by a thin thread. The distance from the centre of the table to the block is 15cm. If the maximum tension the thread can withstand is 5.6N, determine the maximum angular velocity the table can attain before the thread cuts.
 Find the maximum speed with which a car of mass 100kg can take a corner of radius 20m if the coefficient of friction between the road and the tyres is 0.5.
 An object of mass 0.5kg is rotated in a horizontal circle by a string 1m long. The maximum tension in the string before it breaks is 50N. Calculate the greatest number of revolutions per second the object can make.
 An astronaut is trained in a centrifuge that has an arm length of 6m. If the astronaut can stand the acceleration of 9g. What is the maximum number of revolutions per second that the centrifuge can make?
 A small body of 200g revolves uniformly on a horizontal frictionless surface attached by a cord 20cm long to a pin set on the surface. If the body makes two revolutions per second. Find the tension of the cord.
 A circular highway curve on a level ground makes a turn 90^{0}. The highway carries traffic at 120km/h. Knowing that the centripetal force on the vehicle is not to exceed ^{1}/_{10} of its weight, calculate the length of the curve.
 A turntable of record player makes 33 revolutions per minute. What is the linear velocity of a point 0.12m from the center?
 An object 0.5kg on the end of a string is whirled around in a vertical circle of radius 2m, with a speed of 10m/s. What is the maximum tension in the string?
 a) Define centripetal force.
 b) A particle revolves at 4 HZ in a circle of radius 7cm.Calculate its.
 i) Linear Speed.
 ii) Angular velocity. (Take
 c) A 150g mass tied to a string is being whirled in a vertical circle of radius 30cm with uniform speed. At the lowest position, the tension in the string is 9.5N.Calculate,
 i) The speed of the mass.
 ii) The tension in the string when the mass is at the uppermost position of the circular path. (Take g = 10m/s^{2}).
 a) Define angular velocity.
 b) Figure 6 shows an object of mass 2.0 kg whirled in a vertical circle of radius 0.7m at a uniform speed of 50ms^{1}
Figure 6
(i) Determine:
I the centripetal force on the object.
II the tension in the string when the object is at A.
III the tension in the string when the object is at B
(ii) The speed of rotation is gradually increased until the string snaps. At what point is the string likely to snap? Explain
 c) A centrifuge is used to separate cream from milk. A particle of cream has a smaller mass than a particle of milk. Explain how the centrifuge does the separation.
 (a) The figure below shows an object of mass 0.2kg whirled in vertical circle of radius 0.5m at uniform speed of 5m/s
Determine the tension in the string at
(i) Position A
(ii) Position B
(iii) At what point is the string likely to cut. Explain
 (a) Define the term angular velocity
(b) A body moving with uniform angular velocity is found to have covered an angular distance of 170 radians in t seconds. Thirteen seconds later it is found to have covered a total angular distance of 300 radians. Determine t.
(c) The figure below shows a body of mass m attached to the centre of rotating table with a string whose tension can be measured


The tension, T, on the string was measured for various values of angular velocity. The distance r of the body from the centre was maintained at 30cm. the table below shows the results obtained
Angular velocity (rad s^{1})  2.0  3.0  4.0  5.0  6..0 
Tension, T (N)  0.04  0.34  0.76  1.30  1.90 
(Angular velocity)^{2} ^{2} 
 Complete the table above
(ii) Plot the graph of T (yaxis) against ^{2}
 From the graph, determine the mass, m, of the body, given that T = m^{2}r – c, where c is a constant.
 Determine the constant c and suggest what it represents.
 The setup in figure 5 below shows a 50g mass being whirled on a horizontal circular path and balanced by hanging mass M.
Figure 5
Rotich used the above setup to investigate the variation of periodic time, T and the radius, r of the path of the 50g mass and obtained the result shown in the table 1.
Radius, r (m)  0.50  0.41  0.33  0.24  0.16 
Periodic time, T (s)  0.99  0.90  0.81  0.69  0.56 
T^{2 }(s^{2}) 
(a) Complete the table
(b) Use table 1 above to draw the graph of T^{2} against r on the grid provided
(c) From the equation , determine from the graph the value of the force, F that keeps the 50g mass in the horizontal circular path (i.e m = 50g)
 a) Distinguish between centripetal and centrifugal force
 b) The figure shows a motor used by a student in the laboratory to investigate the variation of speed and force on a 10.0 kg mass kept at a fixed distance r from the centre of the rotation
The speed corresponding forces were entered in the table as shown below.
V^{2}(M/S^{2})  0.2  0.4  0.8  1.4  2.2  3.0  3.4  3.6 
F(N)  0.6  1.2  2.4  4.08  6.48  8.90  10.0  10.7 
On the grid provided , plot a graph of F (y axis ) against v^{2 }
 ii) calculate the slope of the graph
iii) Given that F = mv ^{2} find the radius of the rotor
r
 State two factors that affect the centripetal force on a body dencibly circular motion
 a) A body in a uniform circular motion experience acceleration despite having constant Explain (2mk)
 A car travelling with uniform speed on a level circular path is likely to experience skidding experience (2 mk)
 The figures below shows a 40g wooden block being whirled with uniform speed in a horizontal circular path of radius . If it takesseconds to describe an area of length
Figure 6
 Identify the forces acting on the wooden block
 Determine the linear velocity of the block
 Determine the centripetal force
 An object of mass 0.5kg is attached to one end of a light inextensible string and whirled in a vertical circle of radius 1.0m and centre O as in figure 8
If the tension on the string when the object is at the lowest position, A is 13.0N
Calculate;
 The velocity V of the object
 The tension on the string when the object is at the highest point C of the circle
 Explain why a pail of water can be swung in a vertical circle without the water pouring out
 A string of negligible mass has a metal ball tied at the end of the string 100cm long and the ball has a mass of 0.04kg. The ball is swinging horizontally, making 4 revolutions per second.
Determine;
 a) the angular velocity in radian/second
 b) the angular acceleration
 c) The tension on the string
 d) The linear velocity
(b) A centrifuge is used to separate three liquid by rotating it in circular path
Arrange the density of the liquids in descending order (highest to lowest)
 a) Define angular displacement.
 b) A stone having a mass of 1.5 kg is whirled round on the end of a string in a horizontal circle as shown below. The speed of the stone is V m/s and the radius of the circle is 0.8m.
Calculate:
(i) The tension, T in the string given that it makes an angle of 37° with the horizontal
plane of the circle.
(ii) The value, of the velocity, V.
(iii) The periodic time, T, of the motion.
 c) In circular motion, there is acceleration and yet the speed is constant.
Explain.
 a) Explain why the moon is said to be accelerating when revolving around the earth at constant speed
 b) When is a satellite said to be in a “parking orbit”?
 c) A mass of 0.5kg is rotated by a string at a constant speed V in a vertical circle of radius 1m. The maximum tension in the string is 50N.
 i) Indicate on the diagram in figure 11 the positions for the maximum tension(1mk)
Figure 11
 ii) Write an expression for this maximum force experienced
iii) Use your expression to determine the value of V
 iv) Hence determine the minimum tension in the string
 (a) Distinguish between angular and linear velocity.
(b) Explain why bodies in circular motion undergo acceleration even when their speed is constant.
 A particle moving along a circular path of radius 5cm describes an arc length of 2cm every second. Determine:
 Its angular velocity.
 Its periodic time.
 A stone of mass 40g is tied to the end of a string 50cm long and whirled in a vertical circle at 2 revolutions per second. Calculate the maximum tension in the string.
 The figure below shows a container with small holes at the bottom in which wet clothes have been put.
When the container is whirled in air at high speed, it is observed that the clothes dry faster. Explain how the rotation of the container causes the clothes to dry. (3 mks)
 a) Define the term angular velocity
 b) The diagram below shows a spring tied to an object of mass M and made to rotate in a circular path of radius r.
Figure 9
Fig 9
 i) What provides the force that keeps the object moving in the circle
 ii) The speed of the object is constant, why is there acceleration
iii) Although there is a force acting on the object, no work is done on the object. Explain
 iv) If the object is whirled faster and faster, what will happen to the
reading of the spring balance. Give an explanation for your answer
 c) The object used above of mass 0.5kg is then whirled in a vertical circle at 5m/s as shown in the figure below.
Figure 10
Fig 10
Calculate the tension in the string when the object is at the position shown
 Figure 9 shows a pail of water being swung in a vertical circle.
Fig. 9
Explain why the water does not pour out when the pail is at position A as shown.
 a) State two important factors to be considered when setting the banking angle of a road.
 b) A ball of mass 2kg is whirled a the end of a string in a horizontal circular path at a Speed of 5ms^{1}. if the string is 2.0m long find.
 i) the angular velocity of the
 ii) the tension in the string.
(b) In an experiment to investigate the relationship between centripetal force (F) and angular velocity (w) for a body of mass 2 kg moved in a circle of diameter 120cm ,the graph below was drawn
Determine the linear velocity when the centripetal force is 5N.
 c) (i) The figure below shows a ball being whirled in a clockwise direction in a vertical plane.
Sketch on the figure the path followed by the ball if the strings cuts when the ball is at position A.
(ii) A body having uniform motion in a circular path is always accelerating. Explain.
 d) (i) The figure below shows a trolley moving on a circular rail in a vertical plane. Given that the mass of the trolley is 250g and the radius of the rail is 1.5m
(i) Determine the minimum velocity at which trolley passes point X.
(ii) If the trolley moves with a velocity of 4m/s as it passes point Z, Find
Find
(I) angular velocity at this point
(II) The force exerted on the rails at this point.
 (a) A stone of mass 450g is rotated in a vertical circle at 3 revolutions per second. If the string has a length of 1.5m, determine:
(i) The linear velocity.
(ii) The tension of the string at positions A, and B

Fig.2



(b) A stone is whirled with uniform speed in horizontal circle having radius of 10cm. it takes the stone 10 seconds to describe an arc of length 4cm.
(i) The angular velocity ω.
(ii) The period time T.
(c) State two factors affecting centripetal force.
 (a) Define the term angular velocity
(b) The graph below was obtained when an experiment to investigate the variation of the centripetal force, F, with the radius, r of the circle on which a body rotates was performed.
From the graph, determine the angular velocity, w of the body given that m = 100g and
F = mw^{2}r + c where c is a constant.
(c) Explain why the moon is said to be accelerating when revolving around the earth at
constant speed.
(d) Figure 7 below shows a ball whirled in a clockwise direction in a vertical plane.
Fig 7
(i) Sketch on the diagram above (fig. 7), the path followed by the ball if the string cuts when the ball is at position Q.
(ii) State two ways in which the centripetal force on the ball in its circular motion, would be
reduced.
 The figure below shows the path of a stone attached to a string whirled in a space in a horizontal circle.
Sketch on the diagram the path the body follows if the string breaks when the body is at the position shown.
 a) A body moving in a circular path at constant speed is said to be accelerating.
 b) Figure 10 below shows a bucket filled with water moving round in a vertical circular path of radius 1m
Fig 10
If the mass of water is 5kg and the speed of the bucket is 20m/s;
 i) Explain why the water is not falling down when the bucket arrives at point C of the Circular path.
 ii) What is the net force on water at point C.?
iii) Show by calculation that this net force is greater at point A than at point C.
 iv) Calculate the value of the angular velocity
 (a) A stone is whirled in a vertical circle as shown in figure 8 below. A, B,C and D are various positions of the stone in its motion.




Fig. 8

The stone makes 2 revolutions per second in a circle of radius 0.4m, and has a mass of 100g.
(i) Calculate:
I The angular velocity
II The centripetal force
(ii) At C where the stone has acquired a constant angular speed, the string cuts. On the same diagram (figure 8), sketch the path of the stone.
(iii) The stone takes 0.5 seconds to land on the ground. How high is point C above the ground.
(iv) How far does it travel horizontally before hitting the ground.
(b) A bullet of mass 200g traveling at a velocity of 600m/s hits a bag of sand of mass 29.8 kg hanging from a support as shown in figure 9 below. The bullet embeds itself in the sand and the bag swings to the right and rises through a vertical height, h (cm).
Calculate the vertical height, h that the bag and the bullet rises.


 A lead shot of mass 40g is tied to a string of length 70cm. It is swung vertically at 5 revolutions per second.
(a) Determine;
(i)Periodic time
(ii) Angular velocity
(iii) Linear velocity
(iv) Maximum tension in the string.
(b) The figure 5 below shows a container with small holes at the bottom in which wet clothes have been put. When the container is whirled in air at high speed as shown, it is observed that the clothes dry faster. Explain how the rotation of the container causes the clothes to dry faster.

Fig 5




 a) (i) Differentiate between centripetal and centrifugal forces.
(ii) What provides the centrifugal force needed to make a car travel round in a bend of unbanked road?
(b) Below is a diagram of an aircraft of mass 2000kg together with the pilot performing some air maneuvers in a vertical plane.
If the radius of the circular path is 40m and the aircraft is moving at a velocity of 200ms^{1}. Calculate
(i) The external force F_{1} provided by the air at point C.
(ii) The external force F_{2} provided by the air at point B.
(c) (i) Define dynamic lift.
(ii) A horizontal pipe of radius 2.0cm. At one end, gradually increases in size so that its radius is 5 cm at the other end. Water is pumped into the smaller end at a velocity of 8.0ms^{1}. Find the velocity of water at the wider end.
 a) State 2 factors that affect the magnitude of centripetal force of an object moving a long a curved path.
 b) A stone is tied to a light string of length 0.5m if the stone has a mass of 20g and is swung in a vertical circle with a uniform angular velocity of 6 revolutions per second determine:
 i) The period, T
 ii) The tension of the string when the stone is at
(I) bottom of the swing
 II) The top of the swing
III) Linear velocity
 The figure 2 below shows a body of mass 1000Kg which moves along a circular path in vertical plane.
If the radius of the circular path is 10m and the body moves with a velocity of 200ms^{1}, calculate the force which acts on the body at point C.
 a) Define angular velocity.
 b) Figure 6 shows an object of mass 2.0 kg whirled in a vertical circle of radius 0.7m at a uniform speed of 50ms^{1}
Figure 6
(i) Determine:
I the centripetal force on the object.
II the tension in the string when the object is at A.
III the tension in the string when the object is at B
(ii) The speed of rotation is gradually increased until the string snaps. At what point is the string likely to snap? Explain
 c) A centrifuge is used to separate cream from milk. A particle of cream has a smaller mass than a particle of milk. Explain how the centrifuge does the separation.
 (a) The figure below shows an object of mass 0.2kg whirled in vertical circle of radius 0.5m at uniform speed of 5m/s
Determine the tension in the string at
(i) Position A
(ii) Position B
(iii) At what point is the string likely to cut. Explain
 A wooden block mass 200g is placed at distance of 9cm from the centre of a turn table. When the turn table is rotated at a constant angular velocity the block begins to slide off the table. If the frictional force between the block and the turn table is 0.8N, determine:
 a) (i) The linear speed of the block.
(ii) the angular velocity of the turn table.
(c) Explain why a body moving with uniform speed in circular motion is said to accelerate.
(b) An object of mass 0.5kg is attached to one end of a light inextensible string and whirled up in vertical circle of radius 1.0m and centre 0 as shown in figure below such that the lowest point A is at the height of 4 m from the ground.
If the tension on the string when the object is at the lowest point A is 13.0N, calculate.
(i) The velocity V of the object.
(ii) The tension on the string when the object is at the
(i) Highest point C of the circle.
(ii) Point B of the circle.
(iii) If the string was to break when the objects is at the lowest point A of the circle. Sketch on the diagram the path traced by the object until it hits the ground.