# Oscillation of a tethered trolley

##### Class experiment

This experiment related harmonic motion to the mathematical cosine function, using the idea of a spot moving around a circle with an angular velocity, ω .

#### Apparatus and materials

- dynamics trolley
- retort stand bases and rods, 2

- G-clamps, large, 2

- runway for trolley

- expendable steel springs, 6

- newton spring balance, ION

- metre rule

- masses, 100 g and 10 g

Plasticine

some means of recording displacement with time (e.g. video, multiflash photography or ticker timer and tape arrangement)

#### Health & Safety and Technical notes

Adjust the trolley mass so that has a simple value, preferably 10 s-1. If has the same numerical value for everyone in the class, the numerical analysis that follows goes more smoothly. A force constant of ~20 Nm-1 is typical.

#### Procedure

Collecting data

**a** Friction-compensate the trolley runway in the direction the trolley will travel (left to right).

**b** Measure and record the mass of the trolley, *m*.

**c **With all 6 springs in position, measure and record the force constant of the spring, *k*, by pulling the trolley to one side by a measured distance using a spring balance.

**d **Start a recording system that will enable** **you to extract displacement – time data for the first half oscillation once the trolley is moving. Then displace the trolley about 10 cm and release it, timing its oscillations to obtain directly the periodic time, *T*.** **

Data analysis

**e **Plot a displacement-time graph for the first half oscillation of the trolley. Check that the value for *T* calculated from your graph (i.e. twice the time for the half oscillation) agrees with the *T* which you measured directly.

####

Teaching notes

Graphics above show the analysis from a ticker tape.

**1 **Once students have produced their displacement-time graph, it is best to proceed with step-by-step instructions appropriate to the method that they have used for recording their x – t data. Your instructions could be as follows.

Draw a semicircle, centre at the midpoint of the half oscillation (O in the diagram above), with a radius equal to the oscillation amplitude. Along the circle diameter make dots for each position of the trolley during its half oscillation.

Draw a line from each dot perpendicular to this diameter, to intersect the circle. Draw lines joining these intersection points to O.

Measure the angles between these various radii (*f*1, *f*2,*f*2, etc. in the diagram above). [The angles should all be equal, within a degree or so.]

**2** The displacements from the centre at equal time intervals are in proportion to the cosines of the angles which change in equal steps.

‘Mapping’ the time scale of the graph onto an angle scale has definite mathematical meaning. Each value of the time *t* has (over one cycle) just one corresponding angle, much as each point on the ground has a corresponding point on a map of some country. Some of the corresponding values are:

**angle ** 2p p p/2

**time** *t * *T T/2 T/4*

Writing for then gives the harmonic oscillator equation:

**3 **Invite students to picture a spot moving steadily round the circle, as above, through points P, Q, R, S etc. at an appropriate angular velocity, . Ask them to calculate w from their measurements, in degrees per second and then radians per second. With w given in units of rad s-1 they can check that is equal to their measured value of *T*.

*This experiment has yet to undergo a health and safety check.*