B. Dynamics · Dynamics: Forces and Motion in a Plane

Circular Motion in a Vertical Plane: Loops, Hills and Swinging Buckets

B2.6 B2.1About 70 minutes

Learning goals

I can draw free-body diagrams at the top and bottom of a vertical circle and write ΣF(toward centre) = mv²/r at each point.
I can find the normal force (apparent weight) or tension at the top and bottom of a loop.
I can find the minimum speed at the top of a vertical circle, v = √(gr), and explain what happens below it.
I can find the speed at which a car leaves the road at the crest of a hill.

Words to know

vertical circle
Circular motion in a vertical plane, such as a roller-coaster loop, a Ferris wheel or a ball swung on a string over your head.

apparent weight
The support force (normal force or tension) on an object. In a vertical circle it is different at the top and the bottom.

critical speed
The minimum speed at the top of a vertical circle, √(gr), at which gravity alone supplies the centripetal force.

clothoid loop
A teardrop-shaped roller-coaster loop: tighter at the top and wider at the bottom, to keep the forces on riders more even.

Upside down and not falling

At the top of a roller-coaster loop, you are upside down, yet you stay pressed into your seat. A bucket of water swung fast enough over your head does not spill. And at the crest of a steep hill on a country road, your stomach lifts and the car feels light. All of these are circular motion in a vertical plane, where gravity sometimes helps and sometimes fights the centripetal force.

The method: top and bottom

At the top and bottom of a vertical circle, every force is vertical, and "toward the centre" is either straight down (at the top) or straight up (at the bottom). Take toward the centre as positive and write:

ΣF(toward centre) = mv² ÷ r

At the bottom (centre above): the normal force or tension points up (toward the centre, +) and gravity points down (−): FN − mg = mv² ÷ r, so FN = m(g + v² ÷ r). You feel heavier.
At the top, inside the loop (centre below): both the normal force from the seat or track and gravity point down, toward the centre: FN + mg = mv² ÷ r, so FN = m(v² ÷ r − g). You feel lighter.
At the top of a hill (riding on the outside of the circle): the road pushes up (away from the centre, −) and gravity pulls down (+): mg − FN = mv² ÷ r, so FN = m(g − v² ÷ r).

On a real roller coaster or a swung ball, the speed changes around the circle (it is slower at the top). Energy conservation, which you will meet in Work, Energy and Simple Harmonic Motion, links the two speeds; here the speeds are given. A Ferris wheel turning steadily is a true example of uniform circular motion in a vertical plane.

Worked example

Example 1: The roller-coaster loop

A 60 kg rider goes through a circular loop of radius 12 m. Her speed is 14 m/s at the top and 20 m/s at the bottom. Find the force of the seat on her at each point, and compare it with her weight (588 N).

Top (toward the centre is down): FN + mg = mv² ÷ r → FN = 60(14² ÷ 12 − 9.8) = 60(16.33 − 9.8) = 392 N.
Bottom (toward the centre is up): FN − mg = mv² ÷ r → FN = 60(9.8 + 20² ÷ 12) = 60(9.8 + 33.3) = 2588 N.
Ratios: 392 ÷ 588 = 0.67 and 2588 ÷ 588 = 4.4.

Top: 392 N (she feels two-thirds of her weight, pressing up into the seat above her head). Bottom: about 2.6 × 10³ N (4.4 times her weight). That large force at the bottom is one reason modern loops are teardrop (clothoid) shaped: a larger radius at the bottom reduces v²/r there.

Free-body diagrams at the top and bottom of the loop in Example 1. At the top, the seat's force and gravity both point toward the centre. At the bottom, the seat must push up hard enough to cancel gravity and provide the centripetal force.

Worked example

Example 2: The swinging bucket

You swing a bucket of water in a vertical circle of radius 0.90 m (shoulder to the water). What is the minimum speed at the top so the water stays in? About how many swings per minute is that?

At the top, the water needs a net downward force mv² ÷ r. Gravity gives mg; the bucket's bottom can only push down (FN ≥ 0).
Minimum speed: FN = 0, so mg = mv² ÷ r and v = √(gr) = √(9.8 × 0.90) = 2.97 m/s.
f = v ÷ (2πr) = 2.97 ÷ 5.65 = 0.53 Hz, or about 32 swings per minute (if the speed were steady).

About 3.0 m/s, roughly 32 swings per minute. Slower than that, gravity would supply more than the centripetal force needed, and the water would fall out of the circle (onto you). The mass cancels, so it works the same for a full bucket and a half-full one.

Worked example

Example 3: Over the crest of a hill

A car drives over a hill whose crest is a circular arc of radius 30 m. (a) What is the apparent weight of the 70 kg driver at 12 m/s? (b) At what speed does the car lose contact with the road?

The centre is below the car. Toward the centre (down) is positive: mg − FN = mv² ÷ r.
(a) FN = 70(9.8 − 12² ÷ 30) = 70(9.8 − 4.8) = 350 N, about half the driver's weight (686 N).
(b) The car leaves the road when FN = 0: v = √(gr) = √(9.8 × 30) = 17.1 m/s (about 62 km/h).

(a) 350 N (b) 17 m/s. Above that speed, gravity cannot bend the car's path as tightly as the road curves, so the wheels lift off. With no grip, it also cannot steer or brake.

The driver's apparent weight at the crest of the hill in Example 3 falls as speed rises: FN = m(g − v²/r). It reaches zero at √(gr) = 17.1 m/s.

The magic speed √(gr)

The speed √(gr) marks the turning point at the top of a vertical circle. An object on the inside of the circle (a bucket of water, a coaster car, a ball on a string) needs at least √(gr) there; slower than that, the string goes slack or the rider would leave the track, which is why coaster cars have wheels that lock under the rails. An object on the outside of the circle (a car over a hill crest) can go at most √(gr) before it leaves the surface.

Just for fun

Why did the physics teacher swing a bucket of water over her head during the lecture?

To show she could stay cool under pressure. (She practised outside first, just in case.)

Hands-on activity

Feel the forces (two options)

Option 1, the bucket swing (outdoors, with a grown-up). You need: a small plastic bucket or a yogurt container with a sturdy handle (string through two holes) and a cup of water. Stand in an open space away from people and windows. Start with gentle back-and-forth swings, then swing it in a full vertical circle at a brisk, steady pace. Measure your arm-plus-handle radius and count swings in 10 s. Calculate v at the top and compare it with √(gr). Then (carefully) slow down: at roughly what rate does the water start to slosh? Wear clothes that can get wet.

Option 2, a sensor app. Many phones can run a free sensor app (such as phyphox) that records acceleration. As a passenger on a hilly road or on a playground swing, hold the phone flat and record. Find where the reading rises above 9.8 m/s² (bottoms of dips) and where it drops below (crests). Relate each to FN = m(g ± v² ÷ r).

Your turn

A bucket is swung in a vertical circle of radius 0.60 m. What is the minimum speed at the top?

A roller-coaster loop has a radius of 15 m at the top. What is the minimum speed at the top for the car to stay on the track without the lock-under wheels pulling it down?

A 0.30 kg ball on a 1.0 m string moves in a vertical circle. Find the tension at the top when v = 5.0 m/s and at the bottom when v = 7.0 m/s.

An 80 kg driver goes over a hill crest of radius 40 m at 10 m/s. Find the driver's apparent weight.

Why is the tension in a string largest at the bottom of a vertical circle?

A Ferris wheel of radius 10 m turns once every 20 s. Find the apparent weight of a 50 kg rider at the top and at the bottom.

More practice: warm-up

At the bottom of a dip, do you feel heavier or lighter?

At the top of a hill, do you feel heavier or lighter?

At the top of a loop (inside the track), which way does the track push on the car?

Write the critical speed at the top of a vertical circle.

Does the critical speed depend on the mass?

More practice: on your own

A 65 kg rider goes through the bottom of a dip of radius 25 m at 25 m/s. Find the seat's force and compare it with the rider's weight.

A stunt motorcyclist rides around the inside of a spherical cage of radius 4.0 m, going over the top. What is the minimum speed at the top?

A 0.20 kg ball on a 0.50 m string moves at 3.0 m/s at both the top and the bottom (a motor keeps the speed steady). Find the tension at each point.

A Ferris wheel of radius 15 m turns once every 30 s. Find the apparent weight of a 60 kg rider at the top and bottom.

What is the greatest speed at which a car can cross a hill crest of radius 60 m without leaving the road?

Challenge

A string breaks at 40 N. A 0.50 kg ball on 0.80 m of it is swung in a vertical circle. What is the greatest speed it can have at the bottom?

A jet pilot pulls out of a dive at 150 m/s. The pilot can tolerate an apparent weight of 6 times her normal weight. What is the smallest radius for the pull-out (at the bottom of the curve)?

Explain, using forces on the water, why the water stays in a bucket swung over your head fast enough, and why it falls out if you go too slowly.

Check your understanding

Quiz

At the bottom of a vertical circle, the tension in a string holding a ball is:

The minimum speed at the top of a vertical circle of radius 2.5 m is:

A car passes over the crest of a hill. The driver's apparent weight is:

At the top of a roller-coaster loop, the forces on a rider point:

A 50 kg rider at the bottom of a dip of radius 20 m moves at 10 m/s. The seat's force is:

Modern roller-coaster loops are teardrop-shaped mainly to:

For parents and teachers

Big idea: In a vertical circle, gravity is sometimes toward the centre (top) and sometimes away from it (bottom), so the support force changes: riders feel heavier at the bottom and lighter at the top (B2.6).

How to help: Next time you drive over a hill or through a dip (as a passenger), ask your teen to predict where they will feel heavier or lighter, and why.

Common difficulties:

Getting the sign of gravity wrong at the top versus the bottom. Always write "toward the centre is positive" first.
Treating the normal force at the top of a loop as pointing up.
Thinking the water stays in a bucket because of an outward force (it is because gravity alone is not enough to pull it around the circle).

Safety: swing a bucket only outdoors, with small amounts of water, away from people and windows.