JEE Physics: Circular Motion

• PHYSICS JEE (MAIN + ADVANCED) CIRCULAR MOTION

Circular Motion

• When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant, its motion is circular with respect to that point
• The fixed point is the center, and the distance is the radius

Kinematics of Circular Motion

Variables of Motion

• Angular Position:
  • To define angular position in space, an origin and reference line are needed
  • It is the angle between the position vector with respect to the origin and the reference line
  • Depends on the choice of origin and reference line
  • Circular motion is two-dimensional or motion in a plane
  • For a particle P moving in a circle with center O and radius r, the angular position is the angle θ between OP and OX
• Angular Displacement:
  • Definition: the angle through which the position vector of a moving particle rotates within a given time interval
  • Depends on the origin but not the reference line
  • If a point rotates through an angle Δθ in time Δt, then Δθ is the angular displacement
• Important points:
  • Angular displacement is dimensionless
  • SI unit is radian
  • Other units include degree and revolution (2π rad = 360° = 1 rev)
  • Infinitesimally small angular displacement is a vector quantity, but finite angular displacement is a scalar
  • Addition of infinitesimally small angular displacements is commutative, whereas addition of finite angular displacements is not
  • Direction of small angular displacement is determined by the right-hand thumb rule
  • If fingers are directed along the motion, the thumb indicates the direction of angular displacement
• Angular Velocity (ω):
  • Average Angular Velocity:
    • ω_av = (Angular displacement) / (Total time taken)
    • ω_av = (θ₂ - θ₁) / (t₂ - t₁) = Δθ / Δt, where θ₁ and θ₂ are the angular positions at times t₁ and t₂
    • Average angular velocity is a scalar

Instantaneous Angular Velocity

• ω = lim (Δt→0) Δθ/Δt = dθ/dt
• Since infinitesimally small angular displacement dθ is a vector, instantaneous angular velocity ω is also a vector
• Its direction is given by the right-hand thumb rule

Important points:

• Angular velocity has a dimension of [T⁻¹] and the SI unit rad/s
• For a rigid body, all points rotate through the same angle in the same time
• Angular velocity is characteristic of the body as a whole (e.g., Earth's rotation ω)
• If a body makes 'n' rotations in 't' seconds, average angular velocity ω_av = 2πn / t
• If T is the period and 'f' is the frequency of uniform circular motion, ω_av = 2π/T = 2πf

Angular Acceleration (α)

• Average Angular Acceleration:
  • If ω₁ and ω₂ are the instantaneous angular speeds at times t₁ and t₂, the average angular acceleration α_av is defined as:
  • α_av = (ω₂ - ω₁) / (t₂ - t₁) = Δω / Δt
• Instantaneous Angular Acceleration:
  • α = lim (Δt→0) Δω/Δt = dω/dt
  • Since ω = dθ/dt, then α = dω/dt = d²θ/dt²

Important points

• Average and instantaneous angular acceleration are axial vectors with dimension [T⁻²] and unit rad/s²
• If α = 0, motion is uniform

Motion with constant angular velocity

• θ = ω, α = 0

Motion with constant angular acceleration

• Variables:
  • ω₀ = Initial angular velocity
  • ω = Final angular velocity
  • α = Constant angular acceleration
  • θ = Angular displacement
• Analogous to linear motion equations:
  • ω = ω₀ + αt
  • θ = ω₀t + (1/2) αt²
  • ω² = ω₀² + 2αθ
  • θ = (ω + ω₀)/2 * t
  • θₙ = ω + α/2 * (2n - 1)

Relation Between Speed and Angular Velocity

• v = ωr
• Where, v is velocity, ω is angular velocity about center of circular motion, r is the particle's position vector from the center

Relative Angular Velocity

• Velocities are relative, and so is angular velocity
• Absolute angular velocity does not exist
• Angular velocity is defined with respect to the origin (the point from which the position vector of the moving particle is drawn)
• The angular velocity of a particle A with respect to another moving particle B is the rate at which the position vector of A with respect to B rotates at that instant
• Mathematically, ω_AB = (Component of relative velocity of A w.r.t. B, perpendicular to the line separating A and B) / (Separation between A and B) = (v_AB) / r_AB

Important points

• If two particles move on concentric circles with different velocities, the angular velocity of B as observed by A depends on their positions and velocities
• If two particles move on the same circle or different coplanar concentric circles in the same direction with different uniform angular speeds ω₁ and ω₂, the rate of change of angle between OA and OB is dθ/dt = ω₂ - ω₁
• Time taken by one to complete one revolution around O w.r.t. the other: T = 2π / |ω₂ - ω₁| = T₁T₂ / |T₁ - T₂|
• (ω₂ - ω₁) is the rate of change of angle between OA and OB, not the angular velocity of B w.r.t. A

Radial and Tangential Acceleration

• Two types of acceleration in circular motion: tangential and centripetal
• Tangential Acceleration (aₜ):
  • Component of acceleration directed along tangent of the circle
  • Responsible for changing the speed of the particle
  • aₜ = dv/dt = rate of change of speed
  • In vector form: aₜ = α x r
  • If tangential acceleration is in the direction of velocity, speed increases, and if it is opposite to the velocity, speed decreases
• Centripetal Acceleration:
  • Responsible for changing the direction of velocity
  • Always present in circular motion
  • Always variable because it changes in direction
  • Also called radial or normal acceleration

Total Acceleration

• Total acceleration is the vector sum of centripetal and tangential acceleration
• a = a_r + a_t
• a = √(a_r² + a_t²)
• tan θ = a_r / a_t

Important Points

• Differentiation of speed gives tangential acceleration
• Differentiation of velocity gives total acceleration
• dv/dt and |dv|/dt are not the same; the former is the magnitude of the rate of change of velocity (total acceleration), while the latter is the rate of change of speed (tangential acceleration)

Calculation of centripetal acceleration

• Consider a particle moving in a circle with constant speed v.
• The change in velocity Δv between points A and B is VB - VA.
• The magnitude of the change in velocity |Δv| = √(VA² + VB² - 2VA VB cos(θ)) = 2v sin(θ/2).
• Distance travelled between A and B = rθ. Time taken Δt = rθ/v.
• Net acceleration a_net = Δv/Δt = (2v sin(θ/2)) / (rθ/v) = (v²/r) * (sin(θ/2) / (θ/2)).
• As Δt → 0, θ is small, so sin(θ/2) ≈ θ/2. lim (Δt→0) dv/dt = v²/r.
• Net acceleration a_net = v²/r. Speed is constant, so tangential acceleration aₜ = dv/dt = 0

Important Point

• If we derived the formula of centripetal acceleration under condition of constant speed, the same formula is applicable even when speed is variable.

Important point

• In vector form a = ω x v

Dynamics of Circular Motion

• If no force acts on a body, it moves in a straight line with constant speed
• For motion in a circular or curved path, a force must be acting on the body
• If speed is constant, the net force is along the inside normal to the path and is centripetal force
• Centripetal force (F_c) = ma_c = mv²/r = mω²r
• If speed varies, there is also a tangential force acting along the tangent to the path, called the tangential force
• Tangential force (F_t) = Ma_t = M dv/dt = Mαr, where α is angular acceleration

Important Point

• mv²/r is not a force itself
• It is the value of the net force along the inside normal responsible for circular motion
• It could be friction, normal force, tension, spring force, gravity, or a combination

Solving uniform circular motion problems:

• Identify forces along the normal (towards center), calculate resultant, and equate to mv²/r
• In non-uniform circular motion, identify forces along the tangent, calculate their resultant, and equate to mdv/dt or *md|v|*dt

Circular motion in a horizontal plane

Radius of Curvature

• Any curved path can be assumed to be made of infinite circular arcs
• Radius of curvature at a point is the radius of the circular arc fitting the curve at that point
• If R is the radius of the circular arc at a given point P where velocity = v, then centripetal force = mv²/R
• R = mv²/Fc
• Centripetal force Fc = Component of force perpendicular to velocity (F⊥): R = mv²/F⊥ = v²/a⊥
• a is the component of acceleration perpendicular to velocity
• If a particle moves in a trajectory given by y = f(x), then radius of curvature can be determined by the formula provided (Leibniz formula)

Motion in a Vertical Circle

• Point mass tied to a string of length l whirled in a vertical circle
• At an angular position θ, forces acting are tension T and weight mg
• Applying Newton's law along the radial direction: T - mg cos θ = mv²/l
• T = mv²/l + mg cos θ
• The point mass completes the circle only if tension is never zero
• If tension becomes zero at any point, string will slack and motion will be projectile motion
• From equation, tension decreases with increase in θ and v, so tension is minimum at the top
• If tension is momentarily zero at the highest point, the body can still complete the circle

Condition for Completing the Circle (or Looping the Loop)

• Tmin ≥ 0 or Ttop ≥ 0.
• Ttop + mg = mvtop²/l.
• Looping the loop requires Ttop ≥ 0, resulting in minimum velocity at the top: v_top ≥ √(gl)
• Conserving mechanical energy between the lowest and topmost points, (1/2)mu² = (1/2)mvtop + mg(2l)

Relative Speed and Conditions for Tension

-For the velocity at the lowest point must be u ≥ √(5gl). -If the velocity at the lowest point is just adequate for looping the loop, diverse quantities will have specific values

Oscillation or Leaving the Circle

• For non-uniform circular motion in a vertical plane, if velocity at the lowest point is less than √(5gl), the particle will not complete the circle

When speed becomes zero before tension zero,

• the ball never rises above the level of the center and is confined within C and B positions(| θ | < 90°)

when Tension becomes zero before speed zero,

• ball rises above the center and moves beyond point B (θ > 90°)

Condition for Looping the Loop in some other Cases

Cases 1: A mass moving on a smooth vertical circular track. Minimum horizontal velocity at the lowest point = √(5gl) by a calculation similar to the article mentioned earlier i.e, (motion in vertical circle).

Cases 2: A Particle attached to a light rod rotated in vertical circle. Condition for just looping the loop, velocity v = o at highest point. (even if tension is zero, rod will not slack and a compressive force can appear in the rod).

Circular Turning on Roads:

• Vehicles travel along near circular arc when turning
• Turning requires force to produce required centripetal acceleration
• The centripetal force is provided by (i) friction only, (ii) banking of roads only ,and (iii) friction and banking of roads both.

Centrifugal Force

• When rotating in a circular path and centripetal force vanishes, leave circular path.

Effect of Earth’s Rotation on Apparent Weight

• The earth rotates about its axis at on angular speed of one revolution per 24 hours.
• Every point on earth moves in a circle
• Line jointing North and South pole is the axis of rotation.
• The force acting radially outward due to inertia, away from the centre is called centrifugal force