Motion in a vertical circle

The string slackens at the top if v_top < √(gr); complete loops need v_bottom ≥ √(5gr).

Motion in a vertical circle0 min · Free lecture

In this lesson

  • The string slackens at the top if v_top < √(gr); complete loops need v_bottom ≥ √(5gr).

The roller coaster question: how fast at the bottom to survive the top?

At the top, gravity alone can provide the centripetal force: mg = mv²/r → v_min = √(gr). Energy from bottom to top: ½mv_b² = ½mv_t² + 2mgr → v_b,min = √(5gr).

Key results: T_top = m(v²/r − g), T_bottom = m(v²/r + g). Tension is maximum at the bottom, minimum at the top.

vtop,min=gr,vbottom,min=5grv_{top,min} = \sqrt{gr},\quad v_{bottom,min} = \sqrt{5gr}

Vertical circle limits

Worked example

In uniform circular motion, the acceleration of the particle is directed:

  1. Speed is constant but velocity direction changes continuously.
  2. The change in velocity points toward the centre → centripetal acceleration a_c = v²/r toward the centre.

Answer: Toward the centre of the circle

Transcript (0 min)
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Motion in a vertical circle — FemtoLearn.
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The string slackens at the top if v_top < √(gr); complete loops need v_bottom ≥ √(5gr).

Frequently asked

Where is tension maximum?

At the bottom, where gravity and centripetal demand add.

Printable notes (free)

Practice

Free · 4 questions with full solutions
  1. Q1 · Numerical · difficulty 1/5

    A wheel rotates at 270 rpm. What is its angular velocity in rad/s?

  2. Q2 · Numerical · difficulty 1/5

    A point is at distance 1.7 m from the axis of a wheel rotating with angular velocity 11 rad/s. Find its linear speed.

  3. Q3 · Numerical · difficulty 2/5

    A particle moves on a circle of radius 1.5 m with constant speed 6 m/s. Find the magnitude of its centripetal acceleration.

  4. Q4 · MCQ · difficulty 1/5

    In uniform circular motion, the acceleration of the particle is directed:

    • Along the tangent to the circle
    • Toward the centre of the circle
    • Away from the centre of the circle
    • It is zero