# Experimental test of F = mv²/R

##### Class practical

Measuring the variables involved in circular motion, to find out whether the formula for centripetal acceleration, F + mv²/R, is reasonable.

#### Apparatus and materials

For each student pair

Stopwatch or stopclock

Centripetal force kit

Metre rule

#### Health & Safety and Technical notes

Ensure the bungs are securely attached and that there is sufficient space around each student when whirling the bung.

To make a centripetal force kit, as shown in the diagram below, you will need:

• Glass tube with smooth fire-polished rim, 15-20 cm long and 3-4 mm bore. To make this easier to grasp, it can be placed inside a sleeve of tightly-fitting rubber tubing.
• Rubber bung 2-3 cm diameter, 2.5 cm long with a dowel in one of its own holes
• Wire hook
• Small, square piece of card with two holes, used as an indicator*
• Metal washers, up to 15 (200 g)
• Thin string or nylon cord, about 1.5 m long

* The indicator will be adjusted so that it is just below the glass tube when the bung is at the required radius. You can use a paper clip in place of the square indicator, but this is less convenient when rotating.

Pass one end of a piece of cord 1.5 m long first through the holes in a square indicator, then through the glass tube and finally through one hole of the bung. Pass the end back through the other hole of the bung and anchor it by plugging the hole with a short wooden rod (dowelling). Finally, hitch the string round the wooden rod or tie it off. At the other end of the cord, slide the washers which provide the accelerating force, above the wire hook.

#### Procedure

a Hold the glass tube, making sure there is no one too near, and whirl the bung around your head, keeping the indicator just below the glass tube. The indicator will probably rotate slightly when it is not touching the tube; this can be helpful in deciding whether it is clear of the tube or not. While you continue doing this your partner should measure the time for 50 complete orbits. This is best done by counting '3,2,1,0,1,2,3..' with the timer started at 0.

b Repeat step a, with your partner doing the whirling and you the timing. Average both results and work out the orbital period, T, in seconds.

c Measure the radius, R, of the orbit in metres. From T and R, calculate the orbital speed of the bung, v, in metres per second.

d Find the mass of the bung in kg and then calculate the value of the predicted centripetal force in newtons, using mv²/R

e Ignoring friction at the top of the glass tube, the centripetal force is equal to the weight of the washers. Is mv²/R roughly equal to the weight of the washers? Take g = 10 N/kg.

f The force F may be varied by adding more washers.

#### Teaching notes

1 This experiment assumes that students have already been taught that centripetal acceleration,a = v²/R. They are now finding out whether the formula for centripetal acceleration, F = ma = mv²/R, is reasonable. F is the real force (the weight of the washers acting vertically downwards) and mv²/R is the predicted centripetal force for a given orbit.

Students can produce a table of results and in each case the force creating the circular motion, F, the tension in the string, is the weight of the washers. They compare this force with the calculated value of the force which is needed to perform a given orbit, mv²/R

2 There are two difficulties which students commonly have with this experiment:

• using F = ma in the derivation, when the centripetal force is at right angles to the motion and the speed of the bung is constant
• understanding that the weight of the washers provides the centripetal force

3 Students may not be able to whirl the bung around keeping the string horizontal and so the actual radius of the orbit is less. However this does not affect the results because the tension in the string has to be resolved too and both cosine factors cancel out. In any case the cosine is very close to 1.

4 If there is time, get students to explore different variables:

• Vary the force F by adding more washers (up to 20). Look for a relationship between F and v. (They should find that the force is proportional to .)
• vary the mass for a constant radius.

Changing to a different radius complicates the discussion because the radius is involved in calculating the velocity. It is much easier to demonstrate empirically that force is proportional to than to show that F is actually equal to mv²/R because R is involved in calculating v