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.
          Difference between maximum and minimum kinetic energy in vertical circular motion is
          📅CEE MBBS 2025
          Q2.
          Two bodies of masses m₁ and m₂ move in circles of radii r₁ and r₂ respectively. If they complete the circle in equal time, then the ratio of their angular speeds ω₁/ω₂ is:
          📅CEE MBBS 2024a
          Q3.
          For what minimum height \(h\) should a body be released on a smooth inclined plane so that, after reaching the ground, it can revolve in a vertical circle of radius \(R\)?
          📅CEE MBBS 2021
          Q4.
          A body is moving uniformly in a circular path. Which is correct?
          📅CEE MBBS 2021
          Q5.
          Uniform circular motion is motion with:
          📅CEE BDS 2021
          Q6.
          In uniform circular motion, the centripetal acceleration is directed:
          📅BP 2014
          Q7.
          A vehicle moves at \(10\ \mathrm{m/s}\) around a frictionless banked curve of radius \(24\ \mathrm{m}\). Find the ideal banking angle (\(g=9.8\ \mathrm{m/s^2}\)).
          Q8.
          A stone tied to a string moves in a circle at constant speed. Immediately after the string breaks, it moves:
          📅BP 2011
          Q9.
          A correctly banked road has radius \(50\ \mathrm m\). To maintain the same banking angle when design speed is doubled, the new radius must be:
          📅BP 2011
          Q10.
          A motorcyclist must complete a vertical circular loop of radius \(r\). The minimum speed at the lowest point, assuming no energy loss, is:
          📅MOE 2014
          Q11.
          A mass \(m\) moves at constant speed \(v\) in a vertical circle of radius \(r\). The string tension at the bottom is:
          📅MOE 2012
          Q12.
          A bucket of water is rotated at uniform speed in a vertical circle of radius \(4\ \mathrm m\). Find the maximum period for which water does not spill at the top.
          📅MOE 2011
          Q13.
          A cyclist takes a turn of radius \(20\ \mathrm m\) at \(10\ \mathrm{m/s}\). Find the angle of lean from vertical, taking \(g=10\ \mathrm{m/s^2}\).
          📅MOE 2011
          Q14.
          For circular motion of radius \(R\) and frequency \(n\), the centripetal acceleration is:
          📅MOE 2010
          Q15.
          A particle completes 10 revolutions in \(6\ \mathrm s\) on a circle of radius \(1\ \mathrm m\). Its speed is approximately:
          📅MOE 2014
          Q16.
          At a fixed turning radius, if a cyclist doubles speed, the required centripetal force becomes:
          📅KU 2014•BP 2015•BP 2017
          Q17.
          The minimum speed at the bottom of a vertical circle of radius \(r\) needed to complete a full loop with a taut string is:
          📅KU 2013
          Q18.
          A particle moves at \(2\ \mathrm{rev/s}\) on a circle of radius \(0.25\ \mathrm m\). Its centripetal acceleration is:
          Q19.
          A particle moves around a circle of radius \(25\ \mathrm{cm}\) at \(2\ \mathrm{rev/s}\). Its centripetal acceleration is:
          Q20.
          A particle moves along a circle of radius \(r\) at constant speed \(v\). Its tangential acceleration is:
          📅MOE
          Q21.
          A vehicle moves over a convex bridge of radius \(100\ \mathrm m\) with tangential acceleration \(3\ \mathrm{m/s^2}\). Find its total acceleration when speed is \(20\ \mathrm{m/s}\).
          Q22.
          A particle of mass \(m\) moves in a circle of constant radius \(r\) with centripetal acceleration \(a_c=krt\). Find the instantaneous power of the total force.
          Q23.
          A particle’s linear speed doubles while its angular speed halves. Its centripetal acceleration becomes:
          Q24.
          A conical pendulum has bob mass \(0.6\ \mathrm{kg}\), string length \(1\ \mathrm m\) and horizontal circular radius \(0.8\ \mathrm m\). Its tension, for \(g=9.8\ \mathrm{m/s^2}\), is:
          📅MOE Bangladesh
          Q25.
          For uniform circular motion, the angular acceleration is:
          Q26.
          At fixed turning radius, doubling a cyclist’s speed makes the required turning force:
          Q27.
          A stone moves at \(5\ \mathrm{m/s}\) on a circular path of radius \(5\ \mathrm m\). Find the magnitude of average acceleration over half a revolution.
          Q28.
          A \(0.5\ \mathrm m\) string holds a stone moving in a horizontal circle with kinetic energy \(20\ \mathrm J\). The radial force is:
          Q29.
          Uniform circular motion has:
          Q30.
          A bucket of water is swung in a vertical circle of diameter \(2\ \mathrm m\). Find the minimum speed at the highest point to prevent spilling.
          Q31.
          A stone is whirled in a horizontal circle while the string length increases. Which relationship is necessarily true for tensions at two radii?
          📅BPKIHS
          Q32.
          A particle’s linear speed doubles and its angular speed is halved. The centripetal acceleration:
          Q33.
          In non-uniform circular motion, centripetal acceleration compared with tangential acceleration:
          Q34.
          Approximately what angular speed would make an object weightless at Earth’s equator? Use \(g=9.8\ \mathrm{m/s^2}\) and \(R=6.4\times10^6\ \mathrm m\).
          Q35.
          The angular speed increases from \(600\ \mathrm{rpm}\) to \(1200\ \mathrm{rpm}\) in \(10\ \mathrm s\). The angular acceleration is:
          Q36.
          A block slides on the smooth OUTSIDE of a hemisphere of radius \(R\). When it loses contact at speed \(v\), its angular position from vertical satisfies:
          Q37.
          A block released from rest at the top of a smooth hemispherical dome of radius \(R\) loses contact after falling a vertical height:
          Q38.
          A block released from rest at the top of a smooth hemispherical dome leaves its surface at angle \(\theta\) from vertical. Then:
          Q39.
          At the top of a smooth hemisphere of radius \(R\), the minimum horizontal speed for immediate loss of contact is:
          Q40.
          Two vehicles make one revolution in equal periods on circles of radii \(R_1\) and \(R_2\). Their centripetal accelerations satisfy:
          Q41.
          A stone whirls in a vertical circle of radius \(\frac56\ \mathrm m\). If maximum tension is four times minimum tension, find the speed at the highest point (\(g=10\ \mathrm{m/s^2}\)).
          Q42.
          A body starts from rest at height \(h\) on a frictionless slope and enters a vertical loop of radius \(R\). Minimum release height for completing the loop is:
          Q43.
          A coin lies at limiting equilibrium \(4\ \mathrm{cm}\) from the centre of a rotating turntable. If angular speed doubles, the new maximum safe radius is:
          📅BPKIHS
          Q44.
          A particle moves with constant frequency. If the radius of its circular path doubles, its centripetal force becomes:
          Q45.
          A \(1\ \mathrm{kg}\) body moves in a vertical circle of radius \(1\ \mathrm m\). Ignoring losses, the difference between kinetic energies at bottom and top is (\(g=10\ \mathrm{m/s^2}\)):
          Q46.
          A water bucket is rotated in a vertical circle of radius \(4\ \mathrm m\). The maximum period without spilling at the top is approximately:
          Q47.
          A mass is whirled at constant speed \(v\) in a vertical circle of radius \(r\). If \(T_{\max}/T_{\min}=2\), find \(\frac{v^2}{rg}\).
          Q48.
          In circular motion, the angular momentum is \(L\). If kinetic energy halves while angular speed doubles, the new angular momentum is:
          Q49.
          A particle of mass \(m\) completes half a revolution at constant speed \(v\). Magnitude of its change in momentum is:
          📅IOM 2009
          Q50.
          Rail spacing is \(1.5\ \mathrm m\), centre-of-mass height \(2\ \mathrm m\), turn radius \(120\ \mathrm m\). Maximum speed before overturning, neglecting rail banking, is:
          Q51.
          A pendulum bob of length \(L\) at the lowest position is given just enough speed to reach \(60^\circ\) from vertical. Find that speed.
          Q52.
          A stone attached to a string of length \(L\) has speed \(u\) at the bottom of a vertical circle. Find the magnitude of velocity change on reaching a horizontal-string position.
          Q53.
          A pendulum is released from rest at angle \(\alpha\). At a later angle \(\theta<\alpha\), its string tension is:
          📅IOM 2008
          Q54.
          For two particles of equal mass and speed in circular orbits of radii \(r_1\) and \(r_2\), the ratio of centripetal forces \(F_1:F_2\) is:
          📅KU 2009
          Q55.
          The centripetal acceleration of a particle moving at frequency \(n\) on a circle of radius \(R\) is:
          📅MOE 2010
          Q56.
          A \(0.1\ \mathrm{kg}\) mass is whirled at \(10\ \mathrm{m/s}\) in a vertical circle of radius \(1\ \mathrm m\). Tension at the highest point for \(g=10\ \mathrm{m/s^2}\) is:
          📅MOE 2055
          Q57.
          A \(2\ \mathrm{kg}\) mass moves at \(4\ \mathrm{m/s}\) at the top of a vertical circle of radius \(1\ \mathrm m\). Tension at the top for \(g=10\ \mathrm{m/s^2}\) is:
          📅IE 2006
          Q58.
          A \(10\ \mathrm{kg}\) bob swings as a conical pendulum with its string at \(30^\circ\) to vertical. Tension is approximately (\(g=9.8\ \mathrm{m/s^2}\)):
          📅IE 2008
          Q59.
          A pendulum bob of mass \(m\) and length \(L\) is released from rest with the string horizontal. Maximum tension at the lowest position is:
          📅IOM 2015
          Q60.
          At fixed turning radius, doubling a cyclist’s speed changes the required centripetal force to:
          📅IOM 2016
          Q61.
          Centrifugal force is:
          📅IOM 2016
          Q62.
          A particle rotates at \(2\ \mathrm{rev/s}\) in a circle of radius \(25\ \mathrm{cm}\). Its centripetal acceleration is:
          📅IOM 2016
          Q63.
          For a vertical circular loop of radius \(1\ \mathrm m\), the minimum speed at the highest point required to keep the string taut is:
          📅IOM 2017
          Q64.
          A particle of mass \(m\) moves on a circular path of radius \(r\) with angular momentum magnitude \(L\). The centripetal force is:
          📅IOM 2017