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Pendulum and spring

Abstract

In this lab, two systems namely a pendulum and a spring that exhibit simple harmonic motion were studied. The systems were subjected to a mass of known weight and allowed to oscillate. Afterwards, the period of each was determined. In addition, the harmonic properties in the systems and their limiting factors were determined through quantitative calculations.

Introduction

Any system that exhibits back and forth periodic motion (oscillator) is important since most systems are typical oscillators. Several systems exhibit periodic motion and their frequencies depend on the properties of the individual systems. The objective of this lab was limited to investigating

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Its natural frequency can be expressed as: Ѡ =√(k/m) … Equation [1]

The relationship between the angular frequency and Ѡ, and the period T, is given by:

Ѡ =2πf = 2π/T … Equation [2]

The expression for the period in terms of k and m is obtained by combining the two equations as shown in Equation [3]:

T = 2π √(m/k) … Equation [3]

Pendulum:

The restoring force in an oscilating Pendulum does not obey Hooke’s Law.

F = -mgsinɵ … Equation [4]

For the case of a small angle, small angle approximation can be used : sin ɵ ≍ ɵ, In effect, the equation will obey Hooke’s Law as shown in Equation [5] below:

F = -mgsinɵ = -mgx/l … Equation [5]

The mg/l can be the effective k, thus the equation becomes:

T = 2π √(m/k) = 2π√(m/mg/l) = 2π√(l/g) … Equation [6]

Procedure:

Pendulum: The mass, m of the pendulum bob was measured and suspended from a meter-long string. The time, t, taken by the pendulum to swing through, N, of complete periods (T) was measured Step 2 was repeated for the following lengths: 1.0, 0.8, 0.6, 0.4, and 0.2m. Finally, the length l, displacement, d and mass, m were

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The position and velocity data for several oscillations was recorded A smaller amplitude oscillation was used and the amplitude and velocity data recorded until the amplitude decreased to half of its initial value. The position of the mass and the hanger was changed and the last step was repeated once more. The mass of the spring was measured and recorded using the balance scale.

Data/Calculations

Pendulum:

Table 1: Data for the spring recorded in the laboratory

Length of string, L (m) Time for N oscillations (s) Period (s) (T) Measured horizontal distance, d. Theoretical period, T Percent error (%)

1.0 41.29 0.2 +- 0.02 0.72 2.006 +- 0.63 90.02

0.64 32.5 0.2 +- 0.02 0.65 1.605 +- 0.63 87.53

0.48 28.35 0.2 +- 0.02 0.58 1.390 +- 0.63 85.61

0.22 21.07 0.2 +- 0.02 0.46 0.941 +- 0.63 78.74

0.055 13.72 0.2 +- 0.02 0.40 0.470 +- 0.63 57.44

0.055 13.14 0.14 +- 0.02 0.39 0.470 +- 0.63 70.21

Sample calculations:

Theoretical Period, T = 2π√(l/g)= 2×3.142√(1/9.81) = 2.006s

Percentage error = [(Period theo-Period exp)/ Period theo] × 100 = (2.006-0.2)/2.006*100 = 90.29%

Q1a. The uncertainty for the experimental and theoretical values of period, T were estimated by calculating their standard deviations as shown in the excel datasheet