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Question 6: Circular Motion
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Page Introduction to formulae 2
Ord. Level Exam Questions: Manipulation-of-equations type questions 3
Ord. Level Exam Questions: Problem-Solving type questions 5
Answers to Ordinary Level Exam Questions 9
Higher Level Questions 11
Guide to answering Exam Questions: 2008 – 1997 15
************* Marking Schemes / Solutions to be provided separately ***************
Angular Velocity is the rate of change of angle with respect to time. The symbol for angular velocity is ω (pronounced “omega”). Mathematically ω = θ/t
Angular Velocity is measured in radians per second (rad/s), where 2π radians corresponds to a full circle (360^0 ).
A little recap on radians An angle (in radians) is defined as the arc length divided by the radius, i.e. θ = arc length/radius. For a full circle the arc length is 2πr, so in this case θ becomes 2π radians. This allows us to convert from radians to degrees if we need to (2π radians = 360^0 )
Revolutions If the object is carrying out full revolutions, then the time corresponding to one full revolution is called the periodic time (symbol T). Therefore in this case the expression ω = θ/t becomes ω = 2π/T
or
If an object is moving in a circle at constant speed, it is accelerating. This is because while its speed is not changing, its velocity is. Why? Because velocity is defined as speed in a given direction, so if direction is changing, even though speed is not, then the velocity is changing, therefore the object is accelerating.
Now for an object undergoing complete revolutions the frequency (f) of oscillation corresponds to the number of revolutions per second, where f = 1/T e.g. if the periodic time is 0.2 seconds then the frequency will be 5 (Hertz).
Relationship between Linear Speed (v) and Angular Velocity ( ω ) Using the relationship velocity = distance travelled/time taken (and for a full circle the distance travelled is the circumference of a circle = 2πr) ⇒ v = 2πr/T but 2π/T = ω ⇒ v = ωr
Centripetal Force The force - acting in towards the centre - required to keep an object moving in a circle is called Centripetal Force.
Centripetal Acceleration If a body is moving in a circle then its acceleration towards the centre is called Centripetal Acceleration.
Formulae for Centripetal Acceleration and Centripetal Force
but because v = rω we also have
And because F = ma we get and also
a = rω^2
F = mrω^2 r
mv Fc
2
v = r ω
ω = 2π/T
2010 (a) A particle describes a horizontal circle of radius r metres with uniform angular velocity ω radians per second. Its speed and acceleration are 6 m s-1^ and 12 m s-2^ respectively. Find (i) the value of r (ii) the value of ω.
2006 (a) A particle describes a horizontal circle of radius 2 metres with constant angular velocity ω radians per second. The particle completes one revolution every 5 seconds. (i) Show that ω is equal to 2π/5. (ii) Find the speed and acceleration of the particle. Give your answers correct to one place of decimals.
Ordinary Level: Problem Solving (involving a reaction force) 2005 (a) A smooth particle of mass 4 kg is attached to the end of a light inextensible string 50 cm in length. The mass describes a horizontal circle with constant speed 3 m/s on a smooth horizontal table. The centre of the circle is also on the table. (i) Show on a diagram all the forces acting on the particle. (ii) Find the tension in the string.
Solution
(i)
(ii) T = mv^2 /r = 4(3)^2 /0. ⇒ T = 72 N
2006 (b) A conical pendulum consists of a particle of mass 4 kg attached by a light inelastic string of length 2 metres to a fixed point p. The particle describes a horizontal circle of radius r. The centre of the circle is vertically below p. The string makes an angle of 30^0 with the vertical. Find (i) the value of r (ii) the tension in the string (iii)the speed of the particle.
Solution
(i) r = 2 sin 30 = 1 m (ii) T cos 30 = 4g ⇒ T = (80/√3) N (iii) T sin 30 = mv^2 /r (80/√3)(0.5) = 4v^2 / ⇒ v = 2.4 m/s
2008 (b) A hemispherical bowl of diameter 10 cm is fixed to a horizontal surface. A smooth particle of mass 2 kg describes a horizontal circle of radius r cm on the smooth inside surface of the bowl. The plane of the circular motion is 2 cm above the horizontal surface. (i) Find the value of r. (ii) Show on a diagram all the forces acting on the particle. (iii)Find the reaction force between the particle and the surface of the bowl. (iv) Calculate the angular velocity of the particle.
2005 (b) A smooth particle, of mass 4 kg, describes a horizontal circle of radius r cm on a smooth horizontal table with constant speed 1.2 m/s. The particle is connected by means of a light inelastic string to a fixed point o which is 40 cm vertically above the centre of the circle. The length of the string is 50 cm. (i) Find the value of r. (ii) Find the tension in the string. (iii)Find the normal reaction between the particle and the table.
2004 (b) A circus act uses a fixed spherical bowl of inner radius 5 m. A girl and her motorcycle together have a mass of M kg, as shown in the diagram. The girl and her motorcycle describe a horizontal circle of radius r m, with angular velocity ω rad/s, on the inside rough surface of the bowl. The centre of the horizontal circle is 3 m vertically below the centre of the bowl. The coefficient of friction between the motorcycle tyres and the bowl is 3/4. (i) Find the value of r. (ii) Show on a diagram all the forces acting on the mass M. (iii)Find the value of ω, correct to two decimal places.
2003 (b) A smooth particle, of mass 2 kg, describes a horizontal circle of radius 0. metres on a smooth horizontal table with constant angular velocity 3 radians per second. The particle is connected by means of a light inelastic string to a fixed point o which is vertically above the centre of the circle. The length of the string is 1 metre. The inclination of the string to the vertical is α. (i) Find α. (ii) Find the tension in the string. (iii)Show that the normal reaction between the particle and the table is 20 − 9√3 N.
A particle of mass 5 kg describes a horizontal circle of radius 0.7 metres with constant angular velocity ω radians per second on a smooth horizontal table. The particle is connected by means of a light inextensible string to a fixed point o which is vertically above the centre of the circle. The inclination of the string to the vertical is α, where tan α = ½. The tension in the string is T newtons, the normal reaction between the particle and the table is R newtons and R = T√5. (i) Write down the value of sin α and the value of cos α. (ii) Show on a diagram all the forces acting on the particle. (iii)Find the value of T and the value of R. (iv) Find the value of ω.
A smooth particle of mass 2 kg describes a horizontal circle of radius r metres with constant angular velocity ω radians per second on the smooth inside surface of a hemispherical bowl of radius 0.5 metres. The centre of the horizontal circle is 0.4 metres vertically below the centre of the circle formed by the rim of the bowl. The normal reaction between the particle and the bowl makes an angle with the horizontal. α (i) Find the value of r. (ii) Write down the value of cos and of sin. αα (iii)Show on a diagram all the forces acting on the particle. (iv) Find the normal reaction between the particle and the bowl. (v) Find the value of ω.
A particle of mass 20 kg describes a horizontal circle of radius length 12 ½ cm with constant angular velocity of 4 rad/s on a smooth horizontal table. The particle is connected by means of a light inextensible string to a fixed point o which is vertically above the centre of the circle. The inclination of the string to the vertical is θ, where tan θ = 5/12. (i) Show on a diagram all the forces acting on the particle. (ii) Show that the value of the normal reaction between the particle and the table is equal to the value of the tension in the string.
2005 (b) (i) r = 30 cm (ii) T = 32 N (iii)N = 14.4. N
2004 (a) v = 2 m s-
2004 (b) (i) r = 4 m (ii) (iii)= 0.85 rad/sec 2003 (a) (i) (ii) v = 14.14 m s-
2003 (b) (i) = 30^0 (ii) T = 18 N (iii)
2002 (i) sinα = 1/√5, cosα = 2/√ 5 (ii) (iii)T = (50√5)/7 N R = 250/7 N (iv) ω = 10/7 rad/s
2001 (i) r = 0.3 m (ii) cos α = 0.6, sin α = 0. (iii) (iv) R = 25 N (v) ω = 5 rad/s
2000 N = T = 104 N
Higher Level
The first few questions here are straightforward and only a little advanced from Ordinary Level. After that they get tricky
2000 (a) A particle is placed on a horizontal rotating turntable, 10 cm from the centre of rotation. There is a coefficient of friction of 0.4 between the particle and the turntable. If the speed of the turntable is gradually increased, at what angular speed will the particle begin to slide?
2005 (a) A conical pendulum consists of a light inelastic string [pq], fixed at the end p, with a particle attached to the other end q The particle moves uniformly in a horizontal circle whose centre o is vertically below p If (^) po = h, find the period of the motion in terms of h
2008 (b) A and B are two fixed pegs, A is 4 m vertically above B. A mass m kg, connected to A and B by two light inextensible strings of equal length, is describing a horizontal circle with uniform angular velocity ω. For what value of ω will the tension in the upper string be double the tension in the lower string?
2011 (b) A and B are two fixed pegs. A is 4 m vertically above B. A mass m kg, connected to A and B by two light inextensible strings of equal length, l , is describing a horizontal circle with uniform angular velocity ω. Find the value of ω if the ratio of the tensions in the two strings is 11: 9.
1995 (a) A light string [ op ], of length l , is fixed at end o , and is attached at the other end p to a particle which is moving in a horizontal circle whose centre is vertically below and distant h from o.
Prove that the period of the motion is g
h
1995 (b) A particle of mass m , attached to a fixed point by a light inelastic string, describes a circle in a vertical plane. The tension of the string when the particle is at the highest point of the orbit is T 1 and when at the lowest point it is T 2. Prove that T 2 = T 1 + 6 mg
A particle of mass 8 kg is describing a circle, with constant speed v , on a smooth horizontal table. It is connected by a light inextensible string of length 3 m to a point which is 1 m vertically above the centre of the circle. (i) Calculate the tension in the string.
(ii) Show that the particle will remain in contact with the table if v < 8 g
(iii)If the speed of the particle is increased to 9. 1 g , calculate the height at which the particle rotates above
the table.
2012 (b) A particle of mass m kg lies on the top of a smooth fixed sphere of radius 30 cm. The particle is slightly displaced and slides down the sphere. The particle leaves the sphere at B. (i) Find the speed of the particle at B. (ii) The horizontal distance, in metres, of the particle from the centre of the sphere t
seconds after it has left the surface of the sphere is
Find the value of k correct to two places of decimals.
2010 (a) A particle of mass m kg lies on the top of a smooth sphere of radius 2 m. The sphere is fixed on a horizontal table at P. The particle is slightly displaced and slides down the sphere. The particle leaves the sphere at B and strikes the table at Q. Find (i) the speed of the particle at B (ii) the speed of the particle on striking the table at Q.
2004 (a) A particle can move on the smooth outer surface of a fixed sphere of radius r. The particle is released from rest on the smooth surface of the sphere at a height 4r/ above the horizontal plane through the centre o of the sphere. Find, in terms of r, the height above this plane at which the particle leaves the sphere.
2003 (b) A particle of mass m is held at a point p on the surface of a fixed smooth sphere, centre o and radius r. op makes an angle α with the upward vertical. The particle is released from rest. When the particle reaches an arbitrary point q, its speed is v. oq makes an angle β with the upward vertical. (i) Show that v^2 = 2gr (cosα − cos β).
(ii) If cos α =
and if q is the point at which the particle leaves the surface, find the
value of β.
2007 (b) A bead slides on a smooth fixed circular hoop, of radius r , in a vertical plane. The bead is projected with speed √(10 gr) from the highest point c. It impinges upon and coalesces with another bead of equal mass at d. cd is the vertical diameter of the hoop. Show that the combined mass will not reach the point c in the subsequent motion.
1984 (a)
A particle moving on the inside smooth surface of a fixed hollow sphere of internal radius 2 m describes a horizontal circle of radius 1m. Calculate the angular velocity of the particle.
1979 (a) A particle moving at constant speed, is describing a horizontal circle on the inside surface of a smooth sphere of radius r. The centre of the circle is a distance ½ r below the centre of the sphere. Prove that the speed of the particle is (^6) gr 2
1
A hollow right circular cone of semi-vertical angle α where tan α = ¾ is fixed with its axis vertical and vertex downwards. The inner surface of the cone is rough with coefficient of friction ½ and the cone rotates about its axis with uniform angular velocity 7 rad/s. A particle of mass m is placed on the inside surface and rotates with the cone at a vertical height h above the vertex. Calculate the normal reaction of the particle with the inside surface and the height h above the vertex if (i) the particle is about to slide down (ii) the particle is about to slip up.
2006 (b) A hollow cone with its vertex downwards and its axis vertical, revolves about its axis with a constant angular velocity of 4π rad/s. A particle of mass m is placed on the inside rough surface of the cone. The particle remains at rest relative to the cone. The coefficient of friction between the particle and the cone is 1/4. The semi-vertical angle of the cone is 30° and the particle is a distance l m from the vertex of the cone. Find the maximum value of l , correct to two places of decimals.
1982 (b) A small bead of mass m is threaded on a smooth circular wire of radius a , fixed with its plane vertical. The bead is projected from the lowest point of the wire with speed u. Show that the reaction between the bead and the wire, when the radius to the bead makes an angle of 60^0
with the downward vertical is (^)
(^2) g
a
u m
A smooth uniform vertical hoop of radius r and mass M kg stands in a vertical plane on a horizontal surface. The hoop threads two small rings, each of mass m kg. The rings are released from rest at the top of the hoop. (i) When the two rings have each fallen through an angle of θo^ on opposite sides of the hoop, show that the normal force of reaction exerted by the hoop on each ring is mg (3cos θ – 2) N where this force is taken to act in the outward direction from the centre of the hoop.
(ii) Show that the hoop will rise from the table if
M
m >.
2000 (a) Tricky if you haven’t seen it before, but straightforward if you have (and it does come up every so often). Get an expression for the friction force (F = μR) and a separate expression for circular motion ( F = mrω^2 ). Then simply equate to get ω > √(4g).
1997 Full question (i) Forces up = forces down and forces left = forces right. Answer: T = mg/2 sin β (ii) Circular motion; use conservation of energy.KE + PE at top = KE + PE at bottom. The trick here is to take the base line as the horizontal line going through A. This means h1 = -l sin β and h2 = -l sin φ. The second equation in circular motion is Fc = mv^2 /r. As usual with this equation, the tricky part is calculating Fc – in this case it is T – mg sin φ, but it may require a little playing around with angles to verify that. (iii)Quite tricky. First up, you must notice that when the string is cut φ = β, so the equation in part (ii) reduces to T = mg sin β. From part (i) T = mg/2 sin β, so if this tension is halved it becomes T = ( ½ ) mg/2 sin β. Now equate this with the general equation T = mg sin β and solve. Answer: β = 60^0. Told you it was tricky. from xkcd
