Vertical circular motion

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  1. Uniform vertical circular motion

    A 0.315 kg ball attached to the end of a 0.700 m string is swung in a vertical circle so that it has a constant speed of 4.50 m/s. What is the tension in the string at the top and bottom of the circular path?

    1. Minimum speed to keep an object on a vertical circular path

      A 0.245 kg ball is attached to the end of a 0.950 m string and swung in a vertical circle. Find the minimum speed the ball needs at the top of the path to continue travelling in a circle.

      Topic Notes
      In this lesson, we will learn:
      • Solving problems involving vertical circular motion


      • An object moving in a circular path is in circular motion. If the speed of the object is constant, it is uniform circular motion.
      • An object in uniform circular motion does experience acceleration, even though its speed is constant. Remember, acceleration is change in velocity, and a velocity is made up of speed and direction. For the object to move in a circle, the direction of its velocity must change constantly. This change in direction is the acceleration, called centripetal acceleration ("centripetal" means "towards the center"). For an object moving in a circular path, the centripetal acceleration vector is always pointed towards the center of the circle.
      • Like any other type of acceleration, centripetal acceleration is caused by a force (called centripetal force). The centripetal force vector also always points towards the center of the circle.
      • In order for an object to be moving in a circular path, the net force acting on the object must be a centripetal force (a force that always is pointed towards the center). When multiple forces act on an object in circular motion, those forces must add up to a centripetal force. It is important to understand that centripetal force is not a separate force that acts on an object. It is a net force which follows a specific rule: it always points towards the center of the circular path.
        • In a horizontal circular motion problem, any forces acting on the object in the vertical direction must balance so that ΣFvertical=0N\Sigma F_{vertical} = 0N (otherwise the object would accelerate vertically). Only horizontal forces will contribute to the net force causing circular motion.
      Period and Frequency

      T=totaltime#ofrevolutionsT = \frac{total time}{\# of revolutions}

      f=#ofrevolutionstotaltimef = \frac{\# of revolutions}{total time}

      T=1fT = \frac{1}{f}

      T:T: period, in seconds (s)

      f:f: frequency, in hertz (Hz)

      Centripetal Acceleration

      ac=v2r=4π2rT2a_{c} = \frac{v^{2}}{r} = \frac{4\pi^{2}r}{T^{2}}

      ac:a_{c}: centripetal acceleration, in meters per second squared (m/s2)(m/s^{2})

      v:v: velocity, in meters per second (m/s)

      r:r: radius, in meters (m)

      T:T: period, in seconds (s)