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Circular Motion formulae for NEET Circular motion is described as a movement of an object while rotating along a circular path. Circular motion can be either uniform or non-uniform. During uniform circular motion the angular rate of rotation and speed will be constant while during non-uniform motion the rate of rotation keeps changing
1. Average angular velocity : It is the rate of change of angular displacement of a particle in a circular motion. It is measured in rad/s ωavg = Δθ/Δt 2. Instantaneous angular velocity : T he instantaneous rate at which the object rotates in a circular path. ω = lim∆t→0 (∆θ/∆t) = dθ/dt 3. Angular acceleration : It is defined as the rate of change of angular velocity of the rotating particle. It is measured in rad/s^2. α = dω/dt = d^2 θ/dt^2 4. Average angular acceleration: αavg= Δω/Δt 5. Instantaneous angular acceleration: α = dω/dt 6. Relation between speed and angular velocity V = rω 7. Tangential acceleration (rate of change of speed) at = dV/dt = r(dω/dt) =ω(dr/dt) 8. Radial or centripetal acceleration ar= (Velocity)^2 /radius of motion of the object = v^2 /R 9. Normal reaction of the road on a concave bridge N = mgcosθ + mv^2 / 10. Normal reaction on a convex bridge N = mgcosθ – mv^2 / 11. Skidding of the vehicle on a level road: When a vehicle makes a turn on a circular path it requires centripetal force. If friction provides this centripetal force then the vehicle can move in a circular path safely Frictional force ≥mv^2 /r μmg ≥mv^2 /r Vsafe ≤ √μmg
12. Skidding of an object on a rotating platform: To avoid the skidding of an object of mass m at a distance r from the axis of rotation on a rotating platform, the centripetal force must be provided by the force of friction. Centripetal force = Force of friction mω^2 r = μmg ωmax= √μm/r 13. Bending of cyclist tanθ =v^2 /rg 14. Banking of road without friction: Consider a vehicle of mass m moving with a speed v on a banked road of radius r. tanθ = v^2 /rg 15. Banking of road with friction v^2 /rg = (μ + tanθ)/(1- μtanθ) 16. Conical pendulum Tcosθ = mg Tsinθ = mω^2 r Time period = 2π 17. Relation among angular variables ω = ω 0 + αt ω 0 = initial angular velocity ω = final angular velocity
Term (symbol) Meaning squared, end fraction. Period (� T T) Time needed for one revolution. Inversely proportional to frequency. SI units of ssstart text, s, end text. Frequency (� f f) Number of revolutions per second for a rotating object. SI units of 1ss1start fraction, 1, divided by, start text, s, end text, end fraction or Hertz (Hz)Hertz (Hz)start text, H, e, r, t, z, space, left parenthesis, H, z, right parenthesis, end text.
Equations
Equation Symbol breakdown Meaning in words Δ�=Δ��Δ θ = r Δ s delta, theta, equals, start fraction, delta, s, divided by, r, end fraction Δ�Δ θ delta, theta is the rotation angle, Δ�Δ s delta, s is the distance traveled around a circle, and � r r is radius The change in angle (in radians) is the ratio of distance travelled around the circle to the circle’s radius.
Equation Symbol breakdown Meaning in words �ˉ=Δ�Δ� ω ˉ=Δ t Δ θ omega, with, \bar, on top, equals, start fraction, delta, theta, divided by, delta, t, end fraction �ˉ ω ˉomega, with,
bar, on top is the average angular velocity, Δ�Δ θ delta , theta is rotation angle, and Δ�Δ t delta, t is change in time Average angular velocity is proportional to angular displacement and inversely proportional to time. �=�� v = rω v, equals, r, omega � v v is linear speed, � r r is radius, � ω omega i s angular speed. Linear speed is proportional to angular speed times radius � r r. Angular speed is the magnitude of the angular velocity. �=2��=1� T = ω 2 π = f 1 T, equals, start fraction, 2, pi, divided by, omega, end fraction, equals, start fraction, 1, divided � T T is period, � ω omega i s angular speed, and � f f is frequency Period is inversely proportional to angular speed times a factor of 2 � 2 π 2, pi,
Figure 1. Angular velocity vs. linear velocity
Angular speed does not change with
radius
Angular speed � ω omega does not change with radius, but linear speed � v v does. For example, in a marching band line going around a corner, the person on the outside has to take the largest steps to keep in line with everyone else. Therefore, the outside person who travels a greater distance per time, has a greater linear speed than the person closest to the inside. However, the angular speed of every person in the line is the same because they are moving through the same angle in the same amount of time (Figure 2).
Figure 2. Angular speed remains the same regardless of distance from the center, but the linear speed increases proportionally with radius. Image adapted from Wikimedia Commons. Original image from Wikimedia Commons, CC BY-SA 4.
