Energy in Vertical Circular Motion

Imagine a roller coaster looping the loop, or a bucket of water swung overhead in a vertical circle. In both cases, the object must maintain a minimum speed at the top of the loop — otherwise gravity wins and the object (or rider!) falls away from the circular path. This is the central challenge of vertical circular motion: unlike horizontal circular motion, the speed changes continuously because gravitational potential energy is being exchanged with kinetic energy as the object rises and falls around the loop. Energy conservation is the key tool here. As an object moves around a vertical circle of radius r, it gains kinetic energy when descending and loses it when ascending. The total mechanical energy (kinetic plus gravitational potential) remains constant provided no friction acts. This means we can calculate the speed at any point on the loop if we know the speed at any other point — a powerful shortcut that avoids dealing with the varying centripetal force at every angle. The critical point is always the top of the loop, where gravitational potential energy is maximised and speed is minimised. If the speed at the top drops below a certain threshold, the normal contact force from the track (or tension in a string) would need to become negative — physically impossible. Setting this force to zero gives the minimum speed condition, which we can then combine with energy conservation to find the minimum launch speed needed.

Key takeaways
  • Setting up the geometry
  • Applying Newton's second law at the top
  • Using energy conservation to find minimum launch speed
  • Speed at any point on the loop
  • String versus track: an important distinction

Setting up the geometry

Consider an object of mass m moving in a vertical circle of radius r. Take the bottom of the circle as the reference level for gravitational potential energy (h = 0). The centre of the circle is at height r, so the top of the loop is at height h = 2r. Any point at angle θ from the bottom is at height r − r cos θ = r(1 − cos θ).

Applying Newton's second law at the top

At the top of the loop, both gravity (mg downward) and the normal force N (downward, since the track pushes inward toward the centre) provide the centripetal force. Newton's second law gives: mg + N = mv²_top / r. For the minimum speed condition, the object barely maintains contact, so N = 0. This yields v²_min,top = gr, or v_min,top = √(gr). This is the slowest the object can travel at the top without losing contact.

Using energy conservation to find minimum launch speed

Applying conservation of mechanical energy between the bottom (height 0, speed v_bottom) and the top (height 2r, speed v_top): ½mv²_bottom = ½mv²_top + mg(2r). Substituting the minimum condition v²_top = gr: ½mv²_bottom = ½m(gr) + 2mgr = (5/2)mgr. Therefore v²_bottom,min = 5gr, giving v_min,bottom = √(5gr). A roller coaster car must enter the bottom of a loop with at least this speed.

Speed at any point on the loop

More generally, using energy conservation between the bottom and a point at height h = r(1 − cos θ): v² = v²_bottom − 2gr(1 − cos θ). This shows that speed decreases as the object climbs and recovers as it descends. The normal force at any angle can also be found by substituting this expression back into the centripetal force equation, allowing engineers to calculate the forces riders experience throughout a roller coaster loop.

String versus track: an important distinction

For an object on the inside of a track (like a roller coaster), the track can only push (N ≥ 0), so the minimum speed condition at the top is v = √(gr). For an object on a string, the string can only pull (tension T ≥ 0), giving the same minimum condition. However, for an object on the outside of a track (like a ball cresting a hill), the track can only push outward, so the object loses contact when the required centripetal force exceeds the gravitational component — a different scenario altogether.

The Key Equation

vmin⁡,top=grandvmin⁡,bottom=5grv_{\min,\text{top}} = \sqrt{gr} \qquad \text{and} \qquad v_{\min,\text{bottom}} = \sqrt{5gr}
  • v_{\min,\text{top}} — Minimum speed at top (m s⁻¹): The slowest speed at which the object can complete the top of the loop without losing contact with the track or going slack on a string.
  • v_{\min,\text{bottom}} — Minimum speed at bottom (m s⁻¹): The minimum speed required at the bottom of the loop so that the object just barely completes the full circle.
  • g — Gravitational field strength (m s⁻²): The acceleration due to gravity, approximately 9.81 m s⁻² near Earth's surface.
  • r — Radius of the circular loop (m): The radius of the vertical circle being traversed.

Worked Example

Problem. A small toy car of mass 0.050 kg travels on the inside of a frictionless vertical circular loop of radius 0.40 m. (a) Calculate the minimum speed the car must have at the top of the loop to maintain contact with the track. (b) Using energy conservation, find the minimum speed the car must have at the bottom of the loop. (c) Calculate the normal force exerted on the car at the bottom of the loop when travelling at this minimum speed. Take g = 9.81 m s⁻².

vtop≈1.98 m s−1,vbottom≈4.43 m s−1,Nbottom=2.94 Nv_{\text{top}} \approx 1.98\text{ m s}^{-1},\quad v_{\text{bottom}} \approx 4.43\text{ m s}^{-1},\quad N_{\text{bottom}} = 2.94\text{ N}

Answer. Minimum speed at top ≈ 1.98 m s⁻¹; minimum speed at bottom ≈ 4.43 m s⁻¹; normal force at bottom = 2.94 N (which is 6 times the car's weight, consistent with the general result N_bottom = 6mg at minimum conditions).

Figure
Diagram
h=0rrBOTTOMv_bottom = √(5gr)TOPv_top = √(gr)mg (down)N=0 (min)N (up)mg (down)h = 2r↕ 2rKE→PEPE→KE
Origins · 1 minChristiaan Huygens and how this was discovered
CH
Christiaan Huygens
1629–1695 · Dutch

Huygens was the first to derive a correct mathematical treatment of circular motion (1659), showing that an object moving in a circle requires a centripetal ('centre-seeking') force proportional to v²/r. He presented this work in his landmark treatise Horologium Oscillatorium (1673), which also included the laws of the pendulum and the mathematics of evolutes. His analysis of centripetal acceleration laid the essential groundwork for understanding why objects in vertical circular motion need a minimum speed — the very condition we study today. Huygens invented the pendulum clock in 1656, dramatically improving timekeeping accuracy from minutes per day to seconds — a breakthrough that transformed navigation and scientific measurement across Europe.

  • Centripetal Force and Acceleration — The minimum speed condition at the top of a vertical loop is derived directly from applying Newton's second law for centripetal acceleration (v²/r), making this concept the essential prerequisite.
  • Conservation of Mechanical Energy — Energy conservation between the bottom and top of the loop (KE + PE = constant) is the primary method used to find how speed varies around the circle and to derive the minimum launch speed.
  • Gravitational Potential Energy — As the object rises through height 2r from bottom to top, gravitational PE increases by mg(2r), directly determining how much kinetic energy — and therefore speed — is lost.
  • Kinetic Energy — The kinetic energy ½mv² at each point determines the object's speed, and the interplay between kinetic and potential energy drives the changing speed throughout the loop.