Vertical Circular Motion

Vertical circular motion is a special type of circular motion in which the forces acting on an object change as it moves along a vertical loop. Specifically, the normal force varies throughout the motion, directly influencing the net force that provides the necessary centripetal force to sustain the motion. These effects are most pronounced at the top and bottom of the loop, which will be the focus of this analysis.

Use this page to revise the following concepts within vertical circular motion:

Vertical acceleration and the normal force

Have you ever been in an elevator or on a platform that suddenly drops? This momentary feeling of weightlessness occurs because the normal force pushing up on your body decreases. When accelerating upwards, you experience the opposite sensation. You feel heavier because the normal force increases as the platform pushes up on you more forcefully.

Assuming the upward direction is positive, the force diagrams below illustrate how the normal force acting on our bodies influences the sensation of motion:

Three side-by-side diagrams showing a stick figure in an elevator with force vectors. Left: 'At Rest' where F_{net}=0 - two equal vectors, weight (F_g) pointing down and normal force (F_N) pointing up where F_N=F_g. Center: 'Accelerating Up' where F_net>0 - two vectors, normal force (F_N) up is longer than weight (F_g) down, showing F_N>F_g. Right: 'Accelerating Down' where F_net}<0 - two vectors, weight (F_g) down is longer than normal force (F_N) up, showing F_N<f_g. Each case has arrows proportional to force magnitude, and the relative lengths of the arrows visually demonstrate the force comparison.

This relationship between acceleration and normal force is key to understanding the vertical forces involved in vertical circular motion.

Bottom of a loop

At the bottom of a loop during vertical circular motion, an object experiences the following forces:

  • centripetal force upwards
  • normal force upwards
  • weight force downwards

Object at the bottom of a circular loop with velocity vector v pointing horizontally to the right, and three force vectors of arbitrary relative lengths: F_N pointing upward, F_g pointing downward, and F_c pointing upward.

Since the centripetal force is the net force required to maintain circular motion, it can be expressed as the sum of all forces acting on the object at this point. Assuming the upwards is positive:

\[F_c = F_N - F_g\]

The normal force is therefore equal to the sum of the centripetal force and the weight force:

\[\begin{align}F_N &= F_c + F_g  \\ F_N &= \frac{mv^2}{r} + mg\end{align}\]

This explains the "heavier" sensation experienced at this point.

Worked Example

A \(200\mathrm{~kg}\) roller coaster travels at \(20\mathrm{~ms}^{-1}\) at the bottom of a circular dip with a radius of \(20\mathrm{~m}\). Assuming no air resistance and \(g = 9.8\mathrm{~ms}^{-2}\), calculate:

  1. The centripetal force
  2. The normal force from the track

Identify known variables:

\(m = 200\mathrm{~kg}\)

\(v = 20\mathrm{~ms}^{-1}\)

\(r = 12\mathrm{~m}\)

\(g = 9.8\mathrm{~ms}^{-2}\)

Substitute values into the centripetal force equation:

\[\begin{align}F_C &= \frac{mv^2}{r} \\ &= \frac{\left(200\right)\left(20\right)^{2}}{20} \\ &= 4000\mathrm{~N}\end{align}\]

The normal force is the sum of the centripetal force and the weight force (direction from \(g\) is already accounted for):

\[\begin{align}F_N &= F_c + F_g \\ &= F_c + mg\end{align}\]

Substitute values:

\[\begin{align}F_N &= 4000 + \left(200\right)\left(9.8\right) \\ &=5960\mathrm{~N}\end{align}\]


Top of a loop

At the top of a loop during vertical circular motion, an object experiences the following forces:

  • centripetal force downwards
  • normal force downwards
  • weight force downwards

Object at the top of a circular loop with velocity vector v pointing horizontally to the left, and three force vectors of arbitrary relative lengths: F_N pointing downward, F_g pointing downward, and F_c pointing downward.

Since the centripetal force is the net force required to maintain circular motion, it can be expressed as the sum of all forces acting on the object at this point. Assuming the downwards is positive:

\[F_c = F_N + F_g\]

The normal force is therefore equal to the difference of the centripetal force and the weight force:

\[\begin{align}F_N &= F_c - F_g \\ F_N &= \frac{mv^{2}}{r} - g\end{align}\]

This explains the "lighter" sensation experienced at this point.

Worked example

A roller coaster travels through a circular loop of radius \(10\text{ m}\). Assuming no air resistance and \(g = 9.8\text{ ms}^{-2}\), what minimum velocity does the car need at the top of the loop to stay on the track?

Identify the known variables:

\(r = 10\text{ m}\)

\(g = 9.8\text{ ms}^{-2}\)

At the top of the loop the forces acting on an object can be resolved as:

\[F_N = F_c - F_g\]

The normal force must be greater than or equal to zero \(F_N \geq 0\) for the roller coaster to stay on the track. Therefore, the minimum velocity occurs at \(F_N = 0\).

\[\begin{align}0 &= F_c - F_g F_c \\ &= F_g \frac{mv^2}{r} \\ &= mg\end{align}\]

Rearrange for \(v\):

\[v = \sqrt{gr}\]

Substitute values:

\[\begin{align}v &= \sqrt{9.8\ \times\ 10} \\ &\approx 9.90\text{ ms}^{-1}\end{align}\]

Note

The minimum velocity to stay on the track at the top of a loop can be calculated directly as:

\[v = \sqrt{gr}\]

Note that this does not depend on the mass of the object.