1'5CIRCULAR MOTION

1
1.5 CIRCULAR MOTION
1.5.1 Describing Circular Motion
1.5.2 Centripetal Acceleration
1.5.3 Centripetal Force
1.5.4 Examples of Horizontal Circular Motion
1.5.5 Examples of Vertical Circular Motion
2
1.5.1 Describing Circular Motion

• Is the ball undergoing circular motion?
• Is there a net force acting on the ball? If yes,
  what is the direction of it?

3

• Angular Displacement
• The position of a particle in a circular track at
  any time t is specified by its angular
  displacement ?, in , is defined by the equation
• ? Eq.(1)
• where is the radius and is the arc
  length.
• Change the subject of Eq.(1) to s, we get
• s Eq.(2)

4

• Angular Velocity
• The linear speed v of a body moving in a circle
  at an instant refers to its speed along the
  at that instant, i.e.,
• v Eq.(3)
• The angular velocity ? of a body moving in a
  circle at an instant is defined as the of its
  anglar displacement at that instant, i.e.,
• ? Eq.(4)

5

• Differentiating both sides of Eq.(2) with respect
  to t, we get
• Then by Eq.(3) and (4), we have
• Eq.(5)
• Angular Velocity, Period and Frequency
• The time T for a circulating body to swept
  through an angle 2? (one revolution) at an
  angular speed of ?, is
• T Eq.(6)
• If the number of revolutions made in one second
  is f, the angle, in radian, swept in one second,
  ?, is given by
• ? Eq.(7)

6
1.5.2 Centripetal Acceleration

• Magnitude of centripetal acceleration
• Consider a body describing circular motion moves
  from A to B in a very short time interval ?t as
  shown. Its speed v remains almost the same while
  the direction changes. The change in velocity
  can be found by a vector .
• a Eq.(8)

7

• Direction of centripetal acceleration
• The direction of centripetal acceleration is
  of the circle as it can be seen that if ?t
  is so small, A and B nearly , vector ?v is then
  to vA (or vB), i.e., along . That
  is why this acceleration is given the name
  centripetal acceleration.

• Conceptual questions
• Does a body moving uniformly in a circle have
  constant acceleration?
• Is the centripetal acceleration of a body moving
  non-uniformly in a circle still given by the
  expression a ?2r?

8
Examination questions

• 2002-IIA-8
• A particle P is moving in a horizontal circle in
  a clockwise direction as shown (top view). The
  following diagrams show the direction of the
  acceleration a of the particle. Which of them
  is/are possible?
• (1) (2) (3)
• A. (1) only
• B. (3) only
• C. (1) and (2) only
• D. (2) and (3) only

9
1.5.3 Centripetal Force

• Magnitude and direction of centripetal force
• According to Newtons first law, a body
  undergoing circular motion must be acted on by
  to produce the centripetal
  acceleration. This force should be directed
  and is called the centripetal force, which
  causes the body to deviate from the motion
  which it would naturally follow in the absence of
  the force. The magnitude of centripetal force
  according to Newtons second is
• F Eq.(9)

• Example
• A ball attached to a string being swung round
  in a horizontal circle is kept in its track by
  the . It the ball is swung round faster, a
  force is needed, and if it exceeds the maximum
  tension the string can provide, the ball
  continues to travel along the of the
  circle at that point.

10

• Agents providing centripetal force
• Centripetal force is not a new type of
  interaction instead, it is refers to the amount
  of force to substain a circular motion at a
  certain speed.
• The sources providing the centripetal force are
  different in different situations, two examples
  are given below.
• A car riding round a bend
• A bead turning round a smooth track

11

• Experimental verification of centripetal force
• The following set up is used to show that F
  m?2r.
• The of the screw nuts are measured. It is
  the in the string.
• The length l is measured, where l .
• The time t for 50 revolutions of the rubber bung
  is measured, so that the period T , and
  the angular speed ? .

12

• As the rubber bung is whirled round a horizontal
  circle, the string is never horizontal, since the
  always tends to pull the rubber bung .
  An component of the tension is therefore needed
  to balance the , so that

• The centripetal force acting on the rubber bung
  is provided by the component of the tension in
  the string, therefore, if F m?2r, we have
• However, since r , we have
• Eq.(10)

Angular speed and radius
13
Examination questions

• 2000-IIA-11
• A student performing a centripetal force
  experiment whirls a rubber bung attached to one
  end of a string which passes through a glass tube
  with smooth openings, and has a weight W hanging
  at its other end. The weight of the rubber bung
  is much smaller than W. The rubber bung is set
  into a horizontal circular motion with angular
  speed ? while the length of the string beyond the
  upper opening of the glass tube is L and this
  portion of the string makes an angle ? with the
  vertical as shown. Which of the following
  statements is/are correct?
• (1) If L is kept constant, ? will decrease
  with ?.
• (2) If ? is kept constant, L will increase with
  ?.
• (3) When W increases, ? will increase.
• A.  (1) only B.  (2) only C. (3) only
• D.  (1) and (2) only E. (2) and (3) only

14

• Centripetal versus centrifugal force
• Consider a boy B revolving round another boy A at
  a distance r in a circlar path centered at A with
  angular speed ? as shown. According to A, the
  horizontal force acting on B is the
  force acting towards A.

• According to B, however, A seems to be revolving
  round himself with radius r centered at at
  angular speed. However, is there a
  centripetal force acting on A?
• In fact, B is frame, and so if he insists
  to apply Newtons laws of motion, a force
  equals the product of the mass of and the
  of the of B. Therefore, there is an
  apparent centripetal force acting on A in Bs
  frame, which is called the force, as it is
  directed of the circle describing by B.

15

• Simulating gravity in space
• It has been suggested to build a huge disc in
  space that is revolving about a axis at a
  angular speed ?, such that passengers inside the
  disc are all subjected to the forces supplied
  by the of the floor, which give them the
  of weight, just like those on the
  earth.
• In doing so, the acceleration should equal the
  , which gives the required angular
  speed,
• Eq.(11)

16
1.5.4 Examples of Horizontal Circular Motion

• Conical Pendulum
• A conical pendulum consists of a bob of mass m
  supported by a string circulating round a
  circle of radius r with a speed v.
• Horizontal circular motion
• Vertical equilibrium
• The tilting angle ? is governed by

17
Examination questions

• 1990-IIA-6
• A spring of unstretched length 10.0 cm, has one
  end fixed to the ceiling. A mass is suspended at
  the other end, and the extension so produced is
  3.0 cm. When the mass is set to rotate in a
  horizontal circular path the length of the spring
  is 16.0 cm. The angle between the spring and the
  vertical is
• A. 15o
• B. 30o
• C. 45o
• D. 60o
• E. 75o

18

• 2005-IIA-5
•   A horizontal elastic cord of natural length a
  is fixed at one end and a small mass m is
  attached to the other end. The mass rotates
  uniformly in a horizontal circle on a smooth
  horizontal surface. When the mass takes time T
  to complete one revolution, the length of the
  cord is 1.5 a. The angular speed of the mass is
  changed such that the length of the cord becomes
  2a. What is the new time for one revolution?
  (Assume that the elastic cord obeys Hookes law.)
  
• A. B. C. D.

19

• Riding round a bend
• When riding round a bend at speed v, it is the
  at the ground that provides the
  centripetal force.

• Horizontal circular motion
• Vertical equilibrium
• Rotational equilibrium about C.M.
• The tilting angle ? is therefore governed by
• Eq.(12)

20

• Toppling accidents
• During riding round a bend at a definite speed,
  if the rider makes an inclined angle
• - smaller than that given by Eq.(12), the rider
  would topple .
• - greater than that given by Eq.(12), the rider
  would topple .
• Skidding (slipping) accidents
• When the riding speed is too high, the
  centripetal force required may the maximum
  friction that can be supplied, i.e. the
  . If
• the rider will skip for the static
  friction no longer can hold the rider in its
  track.

21
Examination questions

• 1985-IIA-3
• A motorcyclist going round a corner on a level
  road leans over at an angle to the horizontal.
  The reason for this is
• A. to allow his weight to exert a torque about
  the contact point on the ground.
• B. to increase the frictional force between the
  motorcycle and the road.
• C. to lower his centre of mass.
• D. to provide the centripetal force to reduce
  the radius of curvature of his path.
• E. to reduce the radius of curvature of his
  path.

22

• Driving round a bend
• It is the friction between the tyres
  and the ground which provides the centripetal
  force for a car to turn round a bend.

• Horizontal circular motion
• Vertical equilibrium
• Rotational equilibrium about C.M.
• Eliminating f1 and f2, we get
• Solving for N1 and N2, we have
• Eq.(13)

23

• Toppling accidents
• According to Eq.(13), N1 becomes negative when
• in which case the car would topple about the
  wheel.
• Skidding (slipping) accidents
• When the driving speed is too high, the
  centripetal force required may the maximum
  friction that can be supplied, i.e. the
  . If
• the car will skip for the static friction
  no longer can hold the car in its track.

24

• Rounding a banked bend
• To prevent skidding, it is preferable to have the
  centripetal force not rely only on the friction
  of the road. This can be achived by
  the road, so as

• to have the component of the to act
  as the centripetal force.
• Horizontal circular motion
• Vertical equilibrium
• Eliminating the normal reaction N, we get
• Eq.(14)
• Note that for a given radius of bend, the angle
  of banking is only correct for speed.

25
Examination questions

• 1997-IIA-4
•  
• The figure shows a car moving round a corner
  with a radius of 8 m on a banked road of
  inclination 20o. At what speed would there be no
  friction acting on the car along OA?
•  
• A. 5.0 ms-1
• B. 5.2 ms-1
• C. 5.4 ms-1
• D. 5.6 ms-1
• E. 5.8 ms-1

26

• 2000-IIA-8
•  
• A car of mass m is moving with speed v on a
  banked road along a circular path of horizontal
  radius r. The angle of inclination of the road
  is ?. If the centripetal force arises entirely
  from a component of the normal reaction N from
  the road, which of the following relations is
  correct?  
• A. Ncos? mg
• B. N mgcos?
• C. v2 gr/sin?
• D. v2 gr/tan?
• E. N/sin? mv2/r

27

• Aircraft turning in flight
• An aircraft can fly in the sky because of the
  force at angles to the surface of its
  wing. In order to turn,

• the aircraft has to so that the
  componet of the lift supplies the centripetal
  force.
• Horizontal circular motion
• Vertical equilibrium
• Eliminating L, we get
• Eq.(15)
• Note that before turning, L , so immediately
  after banking, the component of L is
  smaller than mg, and the height will be lost,
  unless L is , say, by increasing the during
  turning.

28
Examination questions

• 1995-IIA-6
• An aircraft flies along a horizontal circle of
  radius 10 km with a constant speed of 155 ms-1.
  Calculate the angle between its wings and the
  horizontal.
• A. 11.5o
• B. 12.0o
• C. 12.5o
• D. 13.0o
• E. 13.5o

29

• The rotor
• A rotor found in an amusement park is a rapidly
  spinning upright drum inside which players can
  remain pinned against the wall without falling.

r

• As the rotor spins, the centripetal force on the
  player is supplied by the by the wall N,
  while the weight is balanced by the
  by the wall.
• Horizontal circular motion
• Vertical equilibrium
• If the coefficient of friction between the wall
  and the players cloth is ?, we must have f (
  mg) ? , i.e., the range of rotating speed of
  the drum is found by

30

• Centrifuges
• A centrifuge is a device that separate mattes of
  different in liquids.
• Test tubes containing the mixtures in the
  centrifuges are rotated at speed in a

• circle so that matters finally
  moves towards the centre.
• If the horizontal tube of liquid is rotated, the
  force exerted by the must be greater than
  that by the so that a necessary
  centripetal force is provided. A is
  therefore built up along the liquid.
• For any part of the liquid the force due to the
  difference supplies exactly the centripetal
  force needed, but if the part is replaced by
  matter of smaller density, the force is too
  so the matter moves .

31
1.5.5 Examples of Vertical Circular Motion

• Whirling freely with a rod
• Consider a rod of length r attached to the bob of
  mass m being whirled freely about a smooth joint.
  Assume the speed of the bob at the lowest
  position be vo.
• Vertical circular motion
• Conservation of mechanical energy

• If T gt 0, the rod is in while if T lt 0, the
  rod is in .
• The angle ?max at which the bob reverses
  direction can be found by putting 0 in the
  energy equation. However, if the speed of the
  bob is high enough, the bob will not reverse its
  direction at all.

32
Examination questions

• 1996-IIA-7
• A small object P of mass 0.3 kg is attached to
  one end of a light, rigid rod of length 0.5 m,
  which is free to rotate about the other end O as
  shown. The object is swung to rotate in a
  vertical circle so that it attains a speed of 2 m
  s-1 at its topmost position. What is the force
  exerted on one end of the rod at this instant?
• A. a compressive force of 0.6 N
• B. a tensional force of 0.6 N
• C. a tensional force of 2.4 N
• D. a tensional force of 5.4 N
• E. a compressive force of 5.4 N

33

• Whirling freely with a string
• Consider a rod of length r attached to the bob of
  mass m being whirled freely about a smooth joint.
  Assume the speed of the bob at the lowest
  position be vo.

• Unlike a rigid rod, a string would when
  the tension equals zero.
• Minimum speed at the highest point
• At the minimum speed the centripetal force is
  entirely supplied by the , i.e.,
• Tension at the lowest point

34
Examination questions

• 1993-IIA-8
• The figure shows a small heavy bob P attached to
  a fixed point A on the ceiling by a light
  inextensible string. The bob is pulled aside
  with the string taut and then released from rest.
  Which of the following descriptions is/are true?
• (1) When moving towards the lowest point of its
  path, the angular speed of the bob is
  increasing.
• (2) The centripetal acceleration of the bob is
  constant.
• (3) When the bob is at the lowest point, the
  tension in the string equals the centripetal
  force.
• A. (1) only B. (3) only C. (1)
  and (2) only
• D. (2) and (3) only E. (1), (2) and (3)

35

• Looping the loop
• The loop-the-loop motion on a frictionless
  railway is similar to whirling an object freely
  in a vertical plane by a string, with the tension
  in the string is now being replaced by the
• by the railway.

• The trajectories of the bead for different values
  of vo are shown below

36
Examination questions

• 1991-IIA-6
• The diagram shows a part of the route of a
  roller-coaster in an amusement park. The cart
  descends from H, completes a circular loop A and
  moves to B. If the cart of passengers is to
  complete the central circular track safely, what
  is the minimum velocity of the cart at the bottom
  of the circular track A? (Assume that there is
  no friction between the cart and the track.)
• A. 10m s-1 B. 20 ms-1 C. 22.4 ms-1
• D. 24.5 ms-1 E. 30 ms-1

37

• 1994-IIA-7
• A student whirls a small bucket of water in a
  vertical circle of radius 0.6 m. For no
  spilling, what is the minimum speed of the bucket
  at the highest point of its path.
•  
• A. 2.45 ms-1
• B. 3.46 ms-1
• C. 4.08 ms-1
• D. 4.90 ms-1
• E. 5.77 ms-1

38

• 1995-IIA-5
• A small object of mass 0.05 kg is released form
  rest at the rim of a heavy, smooth semi-spherical
  bowl of radius 10 cm. Find the force acting on
  the object by the bowl when it passes the bottom
  of the bowl.
• A. 0.5 N B. 1.0 N C. 1.5 N
  D. 2.0 N E. 2.5 N
• ball bearing at the highest point A of the loop,
  which of the following graphs correctly shows the
  variation of R with h?
• A. B. C. D.
  E.

• 1997-IIA-5
• A ball bearing is released form rest at a height
  h on a smooth track and completes the circular
  loop of the track. If R is the reaction acting
  on the

39

• 1999-IIA-8
• A small ball-bearing is projected with velocity
  v from the lowest position P of vertical circular
  track which is not smooth. The ball-bearing
  starts to leave the track at Q. Which of the
  following diagrams best represents all the forces
  on the ball-bearing at Q?
• A. B. C.
• D. E.

40

• 2003-IIA-7
• A simple pendulum is pulled horizontal and hen
  released from rest with the string taut. Which
  of the following statements about the tension in
  the string is not correct when the pendulum
  reaches its vertical position?
• A. The tension equals the weight of the pendulum
  bob in magnitude.
• B. The tension attains its greatest value.
• C. The tension does not depend on the length of
  the pendulum.
• D. The tension depends on the mass of the
  pendulum bob.

41

• 2005-IIA-2
• The figure shows a pendulum bob swinging between
  positions x and y.
• Which of the arrows P, Q, R, S represents the
  direction of the resultant force acting on the
  bob when the bob is at position z? (Neglect air
  resistance.)
• A. P B. Q C. R D. S