REPORT FOR LAB A10: MECHANICAL OSCILLATIONS

Nathaniel Kan11/14/01

ABSTRACT:

In this lab we studied the relationship the harmonic motion of a spring-mass system when energy is conserved, when the system is damped and when a sinusoidal force drives the damped system. We then compared the actual results with the theoretical values.

PURPOSE:

See A10 Instruction Sheet page 1.

PRINCIPLES:

Part 1:

We calculate spring constant k with the equation:

K = g / (slope of y vs m)

And effective mass:

M = mindicator + mspring / 3

Part 2:

For damped motion, we have the equation:

My” = - ky – by’

We get the damping constan and the damping periodt:

Sigma = b / 2MOmegadamped = (Omega0

2 – Sigma2)1/2

Part 3:

We find the Quality factor, Q, of a resonant system:

Q = Omega0 / 2Sigma

PROCEDURE:

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We followed the procedure according to the A10 Instruction Sheet.

CALCULATIONS:

See Lab book page 38 for calculation of the spring constant.

K = g/slope = 9.8/1.472 = 6.658 kg/s2

See Lab book page 38 for calculation of the effective mass.

M = 50.9 + 2.5/3 = 51.73 g

See attached graph for calculation of sigma for damped oscillations. Sigma equals the negative slope of the graph of T vs ln(A).

See Lab book page 40 and 41 for calculations of Q.

Q = 7.06 (1/s) / 2(0.1677 mm/s) = 21.05 mm

RESULTS:

Part 1:

We calculated the spring constant at 6.658 kg/s2 (see Lab book page 38). We then measured the period as 0.89 ± 0.05 s. The theoretical period we calculated as 1.17 s. Although our theoretical period does not exactly match our measured period, this does not mean that the theory is wrong. We have not taken into account air resistance, or effective mass. We measured the mass of the spring, and then used that to calculate the effective mass at 51.73 g (see Lab book page 38).

We can see that because Omega = (k/M)1/2 and T = 2(Pi)/Omega, when the mass increases, Omega decreases, and thus the period increases. A higher k means that Omega is smaller, and so the period increases.

Part 2:

See Lab book page 39 for calculations of Sigma, Omegad, and Td. We can see that because Omegadamped = (Omega0

2 – Sigma2)1/2 and because Sigma is much less than Omega0, Omegadamped and Omega0 are almost the same value. Because of this, T and Td are practically the same value.

Part 3:

See Lab book page 40 and 41 for calculations of Q. With damping magnets at distance d2, we find the resonance frequency is about 1.81 ± .01 Hz. The amplitude at resonance is 80.0 ± 0.1 mm. This agrees with our theoretical value of Q.

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