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Definition: Moving in a perfect circle, while velocity has a constant magnitude but changing direction. 

Quantities

  • Angular displacement (θ): Angle through which the object moves.

    • Measured in degrees (º) or radians. 2π radians = 360º​​.

  • Angular speed (ω): Δθ/Δt​.

  • Period (T): time taken to complete one revolution.

  • Link between linear and circular quantities: s = θr and v =​ ωr, where r is the radius.

Centripetal acceleration (ac)

Object moving in a circle:

  • Equation: ac = Δv/Δt = vΔ/Δt = vω = v^2/r = 4rπ^2/T^2.

  • Reason: Since the velocity is changing the direction when an object moves in a circle, there must be an acceleration.

  • Direction: Always directed towards the center of the circle. It generates the centripetal force, which is also always directed towards the center.

 

Centripetal force (Fc) 

  • Equation: Fc = mac = (mv^2)/r = mrω^2. No work, as F is perpendicular to v!

Cases:

  • Satellites in orbit: Centripetal force = Gravitational force, towards the planet's center of mass.

  • Rotor ride: Centripetal force = Gravitational force, towards the planet's center of mass.

    • In this case, Weight force = Friction force.​

  • Turning on a horizontal road: Centripetal force = Friction acting between the tyres and the road.​

    • When skidding:​ (mv^2)/r = μdmg.

    • So that it does not skid: (mv^2)/r < μsmg.

  • Banking on the road: Road banked at an angle θ. Centripetal force = Normal force x sinθ.​​

    • Angle proportional to speed.

    • Examples: cars, cycle velodrome, commercial airline pilots, high-speed trains.

  • Vertical circle with strings: Weight force and string tension must be taken into account.

    • At the top: Fc = Tdown + mg.

      • To keep on moving: v^2 = gr.

    • At the bottom: ​Fc = Tup - mg.

      • Maximum tension on the bottom so ​

                          that it does not break:

                          Tbreak > (mv^2)/r + mg

  • Car on speed bump: Car loses contact when Centripetal force = Weight force, i.e. N = 0.

Newton's law of gravitation:

  • Equation: Gravitational force = Fg = GM1M2/r^2, where M1 and M2 are two different objects' masses, r is the distance between them and G the universal gravitational constant, which is equal to 6.67 x 10^-11 Nm^2kg^-2

  • Direction: Gravitational force is always attractive and acts on every body with mass.

  • Point mass: For a spherical body with uniform density, the entire mass may be assumed to be concentrated at its center, and so the body may be assumed to be a point mass.

Gravitational field strength (g)

  • Definition: "Gravitational field strength is the gravitational force per unit mass experienced by a small point mass placed at a certain point".

  • It is a vector.

  • Equation: g = F/m =GM/r^2, where M is the Mass influencing the small point mass.

    • Units: ms^-2 or GM/r^2.

    • At Earth g = 9.81 ms^-2.

  • Field lines: Towards body's center of mass.

Tip: Why passengers feel as if they are being thrown outwards when in turning on a horizontal road?

"Passengers are in a rotating frame of reference Seen from above the passengers would move in a straight line, according do Newton's First Law of Motion, but friction acts at the seat to provide centripental force to centre of circle. Passengers interpret the reaction to this force as being flung outwards". (Tsokos, 2014).

Topic 6: Circular Motion and Gravitation

6.1 Circular Motion

6.2 Newton's Law of Gravitation