8Circular motion

📚
CIRCULAR MOTION
▢ Centripetal Acceleration and Force:

Table 1: Basic Formulae

Quantity
Formula
Linear velocity
\(v=\omega r\)
Centripetal acceleration
\(a_c=\omega^2r=\frac{v^2}{r}\)
Centripetal force
\(F_c=m\omega^2r=\frac{mv^2}{r}\)
Direction of centripetal acceleration
Towards centre
Direction of velocity
Tangential
▢ Vehicle on Circular Track:

Table 1: Vehicle Turning Formulae

Condition
Formula
Safe turning / no skidding
\(\frac{v^2}{rg}=\mu\)
Banking / inclination relation
\(\frac{v^2}{rg}=\tan\theta\)
Overturning condition
\(\frac{v^2}{rg}=\frac{a}{h}\)
\(a\)
Semi-distance between inner and outer wheel
\(h\)
Height of centre of gravity of vehicle
Critical speed
\(v=\sqrt{\mu rg}\)
▢ Conical Pendulum:
❖ Definition: Mass tied to string and moving in horizontal circle
❖ Symbols:

Table 1: Symbols

Symbol
Meaning
\(O\)
Centre of suspension
\(C\)
Centre of oscillation
\(l\)
Length of string
\(R\)
Radius of horizontal circle
\(h\)
Vertical height = \(l\cos\theta\)
\(\theta\)
Angle made by string with vertical
❖ Equations:

Table 1: Conical Pendulum Formulae

Quantity
Formula
Horizontal component of tension
\(T\sin\theta=\frac{mv^2}{R}\)
Vertical component of tension
\(T\cos\theta=mg\)
Radius
\(R=l\sin\theta\)
Height
\(h=l\cos\theta\)
Speed relation
\(\frac{v^2}{Rg}=\tan\theta\)
Angular velocity
\(\omega=\frac{v}{R}=\sqrt{\frac{g\tan\theta}{R}}=\sqrt{\frac{g}{l\cos\theta}}\)
Time period
\(T_p=2\pi\sqrt{\frac{R}{g\tan\theta}}=2\pi\sqrt{\frac{l\cos\theta}{g}}=2\pi\sqrt{\frac{h}{g}}\)
❖ Special Points:
  • •
    \(\theta\) increases as speed increases
  • •
    For string to become completely horizontal, \(\theta=90^\circ\)
  • •
    At \(\theta=90^\circ\), speed should be infinite and time period zero
  • •
    Hence body cannot be whirled in a perfectly horizontal circle by string alone
▢ Vertical Circular Motion:
❖ Mass Tied to String:
◉ Tension Formula: \(T=\frac{mv^2}{r}+mg\cos\theta\)
◉ Note: \(\theta\) is measured from lowest point
◉ _*table:
    ❖ Critical Velocity:
    ◉ **table:
      ▢ Convex and Concave Roads:

      Table 1: Vehicle on Curved Road

      Case
      Equation
      At special point
      Convex circular road
      \(mg\cos\theta-N=\frac{mv^2}{R}\)
      At highest point: \(N=mg-\frac{mv^2}{R}\)
      Concave circular road
      \(N-mg\cos\theta=\frac{mv^2}{R}\)
      At lowest point: \(N=mg+\frac{mv^2}{R}\)
      ❖ Note: Normal reaction first increases then decreases on both convex bridge and concave road
      ▢ Read and Digest:
      ❖ **table:
        ▢ Two Vehicles on Circular Tracks:
        ❖ Equal Time Period:

        Table 1: \(T_1=T_2\)

        Ratio
        Value
        \(\omega_1:\omega_2\)
        1:1
        \(v_1:v_2\)
        \(r_1:r_2\)
        \(a_1:a_2\)
        \(r_1:r_2\)
        ◉ Reason: \(\omega=\frac{2\pi}{T}\), \(v=\omega r\), \(a=\omega^2r\)
        ❖ Equal Speed:

        Table 1: \(v_1=v_2\)

        Ratio
        Value
        \(\omega_1:\omega_2\)
        \(r_2:r_1\)
        \(a_1:a_2\)
        \(r_2:r_1\)
        ◉ Reason: \(\omega=\frac{v}{r}\), \(a=\frac{v^2}{r}\)
        ▢ High-Yield Recall:
        ❖ **table:
          Q1.
          A body is moving with uniform velocity in circular path then, centripetal acceleration is:
          📅BP 2014
          Q2.
          A four wheeler is moving in a banked track of radius 24 m with velocity 10 m/s. What is the minimum angle of banking so that it won't overturn?
          Q3.
          A stone of mass 'm' is tied to a string of length 'I' and rotated in a circle at constant speed 'v'. If the string is released the stone flies. [BP 2011]
          📅BP 2011
          Q4.
          A road of 50 m radius is banked at correct angle for a given speed. If the speed is to be doubled, keeping the same banking angle, the radius of the road should be changed [BP 201 1]
          📅BP 2011
          Q5.
          "In a death well", a motorcyclist performs his race on circular path of radius (r) then minimum velocity at lowest point is: [MOE 2014]
          📅MOE2014
          Q6.
          A body of mass m is moving in a vertical circle with constant speed of V. The tension on the mass at the bottom of the. circle is [MOE 2012]
          📅MOE 2012
          Q7.
          A can filled with water is revolved in a vertical circle of radius 4 m so that water does not spill down. The maximum period of revolution will be:
          📅MOE 2011
          Q8.
          A cyclist moving with velocity 10 m/s in radius 20 m then angle of inclination of cycle [MOE 2011]
          📅MOE 2011
          Q9.
          A particle of mass m is moving along a circular path of radius R with a frequency and period T. It's centripetal acceleration can be expressed as: [MOE 2010]
          📅MOE 2010
          Q10.
          The radius of circular path is 1 m. A particle makes 10 revolutions in 6 sec. The linear velocity of the particle is: [MOE 2014]
          📅MOE 2014
          Q11.
          A cyclist turns around a curve at 20 km/hr. If he turns at the double of this speed, the tendency to overturn is [KU 2014/ BP 2015/ 2017]
          📅KU 2014•BP 2015•BP 2017
          Q12.
          If a ball is attached to a string and whirled in a vertical circle then the velocity at the lowest point is :
          📅KU 2013
          Q13.
          A body is describing circular motion with constant speed v along circular path of radius r. Then its
          Q14.
          A particle moves in a circle of a radius 25 cm at 2 rev/sec. The acceleration of the particle in m/s' is
          Q15.
          moving along convex bridge of radius of curvature 100m is speeding at 3m/s'. What is its acceleration when its velocity is 20m/s?
          Q16.
          The speed of revolution of a particle going around a circle is doubled and its angular speed is halved. What happens to the centripetal acceleration?
          Q17.
          A 600g object is tied to a string Im long and it is rotated in a horizontal circle of radius 0.8m. Thus the tension produced on the string is [MOE Bangladesh]
          📅MOE Bangladesh
          Q18.
          A particle is moving in a circle of radius with a uniform speed. Its angular acceleration is,
          Q19.
          A cyclist turns around a curve at 20 km/hr. If he turns at the double of this speed, the tendency to overturn is
          Q20.
          A stone tied to one end of string, 5 m long is whirled in horizontal circle at constant speed 5m/s. What is average acceleration of the stone in half revolution?
          Q21.
          A stone tied to one end of light string 50 cm long, is whirled in horizontal circle with constant K.E. of 20J. Then, tension in the string will be,
          Q22.
          Uniform circular motion is motion with
          Q23.
          A man swings a bucket of water in a vertical circle of diameter 2m. The minimum speed with which he must swing the bucket so that the water does not spill is
          Q24.
          A stone tied to one end of a string is whirled in horizontal circle so that length increases gradually. If T, and T, are the tensions in the string when its length be / [BPKIHS]
          📅BPKIHS
          Q25.
          The speed of revolution of a particle going around a circle is doubled and its angular speed is halved. What happens to the centripetal acceleration?
          Q26.
          For a particle in circular motion, the centripetal acceleration is
          Q27.
          What should be the angular velocity of earth so that a body on its equator is weightless
          Q28.
          The angular speed of a car increases from 600 rpm to 1200 rpm in 10 sec. What is the angular acceleration of the car?
          Q29.
          A block released at the top of a smooth hemispherical bowl of radius R slip tangentially with radius vector makes angle 0 with vertical, when its velocity is v. Then angle 0 is,
          Q30.
          A block released at the top of a smooth hemispherical bowl of radius R slips tangentially after falling a vertical height h. Then,
          Q31.
          A block released at the top of a smooth hemispherical bowl of radius R slips tangentially after falling a vertical heighth where radius makes angle 0 with vertical Then 0 is
          Q32.
          What minimum horizontal speed imparted to the block at the top of smooth hemisphere of radius R so that it slips instantly?
          Q33.
          A stone tied to one end of a string of length 5/6m is whirled in a vertical circle so that maximum tension in the string is 4 times the minimum tension in it. The speed of the stone at highest point is,
          Q34.
          For what minimum height h, a body is released on a smooth inclined plane so that it revolves in vertical circle of radius R after reaching the ground
          Q35.
          A horizontal turn table is rotating steadily at angular speed c with a coin on its distance 4cm from the centre o the table. If angular speed of the tu is made 2w then distance of the coin on the table in equilibrium condition is
          📅BPKIHS
          Q36.
          The radius of the circular path of a particle is doubled but its frequency of rotation remains constant. If the initial centripetal force be F, then final centripetal force will be
          Q37.
          A body of mass 1kg is rotating in a vertical circle of radius 1m. What will be th difference in its kinetic energy at the top and bottom of the circle?
          Q38.
          A bucket of filled with water is revolved in a vertical circle of radius 4cm. If g = 10m/s', the time period of revolution will be nearest
          Q39.
          An object is tied to string and rotated in a vertical circle of radius r. Constant speed is maintained along the trajectory. If Tmax/Tmin = 2, then v/(rg) is
          Q40.
          A particle is making uniform circular motion with angular momentum L. If its kinetic energy is made half and angular frequency be doubled, its new angular momentum will be
          Q41.
          If an object of mass 'm' moving in circular path-with uniform speed 'v' which of th following change occurs in half circle [IOM 2009]
          📅IOM 2009
          Q42.
          The distance between two rails is 1.5m. The centre of gravity of the train is at height of 2m from the ground. The maximum speed of the train on a circular path of radius 120m can be:
          Q43.
          A bob of a simple pendulum of length L is at its lowest point. What minimum velocity is imparted to the bob so that the string makes a maximum angle of 60 with vertical.
          Q44.
          A stone tied to a string of length L is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is in the lowest position and has a speed u. The magnitude of change in velocity as it reaches a point where the string is horizontal is
          Q45.
          A particle of mass m is tied to one end of a long in extensible string whose other end is fixed to a rigid support. The particle is taken one side so that the angular displacement from vertical is of The particle is released so that the angular displacement of any instant is 0 (< <). Then the tension in the string must [IOM 2008]
          📅IOM 2008
          Q46.
          48 Two particles of equal masses are revolving in circular paths of radii r, and Tr respectively with same speed. What will IKU 2009 be the ratio of their forces
          📅KU 2009
          Q47.
          A particle of mass m is moving along a circular path of radius R with a frequency n and period T. Its centripetal acceleration can be expressed as [MOE 2010)
          📅MOE 2010
          Q48.
          A body of mass 0.1kg tied by a string is rotating round a vertical circle with speed of 10ms" of radius Im. What is the tension experienced by the string at the highest point: [MOE 2055]
          📅MOE 2055
          Q49.
          A body of mass 2kg is attached to one end of string of length Im & rotated in vertical circle such that velocity at highest point is 4m/s. Here tension is [IE-06]
          📅IE-06
          Q50.
          If an object of mass 10kg is whirled round a horizontal circle of radius 4m by revolving string inclined at 30" to vertical If the uniform speed of the object is 5ms", the tension of the string (approx) is [ IE-08]
          📅IE-08
          Q51.
          When a pendulum is displaced through 90%, the length / and mass m of the pendulum. The minimum strength of string that can withstand tension at the mean position is [IOM 2015]
          📅IOM 2015
          Q52.
          A cyclist turned over at 15 miles/hr. It doubled the speed of cycle then what is the chance of overturning? [IOM 2016)
          📅IOM 2016
          Q53.
          Centrifugal force is
          📅IOM 2016
          Q54.
          A circular object of radius 25 cm, is revolving with a speed of 2 rev/sec. Then what is the acceleration produced? [IOM 2016]
          📅IOM 2016
          Q55.
          For vertical circular motion, radius is Im then its critical velocity is :
          📅IOM 2017
          Q56.
          The mas of particle is 'm' is revolving around a circle having radius r & angular momentum L. Then its centripetal force will be:
          📅IOM 2017