Simple Harmonic Motion and Pendulums SP211: Physics I Fall 2018 Name:

1 Introduction

When an object is oscillating, the displacement of that object varies sinusoidally with time. Simple Harmonic Motion describes this oscillatory motion where the displacement, velocity and acceleration are sinusoidal. The objects we are most interested in today are the physical pendulum, simple pendulum and a spring oscillator.

We can model this oscillatory system using a spring. If we have a spring on the horizontal (one-dimensional motion), and using Hooke’s Law (F = -kx) we can use the following expression: −k ∗ x = m ∗ a (1) In order to solve this expression, we will rewrite it.

−k d2 x(t) = x(t) (2) m dt The only solution that can satisfy this equation is a sinusoidal function. For today, we will be using:

xmaxcos(ω ∗ t + φ) (3) r 2π k ω = = (4) T m In these equations, we have xmax is the Amplitude, so the maximum position in the positive or negative direction. ω is our angular frequency, and the portion ω ∗ t + φ is the phase of our wave. The φ is a phase constant, which describes the position of the wave in a cycle at t = 0 s. If we think about the conservation of Energy with a spring in motion, and we ignore heat and external forces, we can state that: 1 1 E = kx2 + mv2 (5) mechanical 2 2 If we set the initial equation when x = xmax (so v = 0) and the final when x = 0 (so v = vmax), we get the following expression: 1 1 kx2 = mv2 (6) 2 max 2 max 2 k Since we know ω = m , we can reduce this expression to:

vmax = ωxmax (7)

2 Procedure: Simple Harmonic Motion of a Spring 2.1 Find the Mass of Spring and Rubber Bob We won’t need this information immediately, but start the lab by measuring the mass of the spring and the rubber bob. Mass of Spring = kg Mass of Rubber Bob = kg

1 2.2 Measure the Spring Constant Similar to Lab 4, we will start by measuring the spring constant k. For this part of the lab, attach the spring to the hanger, and measure the length of the spring without any mass attached to it. This is your x0, and record it in the appropriate column in the chart below for all masses. Data Table Mass Weight (m*g) Equilibrium Legnth Spring Length (x) x-x0 (m) (x0)

Use the graph below to plot the Weight (y-axis) versus the x-x0 (x-axis). Draw a line of best fit, and then calculate the slope by making a 90◦ triangle. This slope is your Spring Constant.

k = N/m

2 2.3 Measure the Period of the Oscillations Attach the rubber bob to the end of the spring. Pull the rubber bob enough so you can measure oscillations for 10 periods, but not so much that it moves randomly. The goal here is to only have the spring move in the y-direction. Measure 10 periods with your timer and then divide by 10 to get the period for 1 oscillation. Data Table Trial Time for 10 periods (seconds) Time for 1 period (seconds) 1

2

3

AVERAGE

2.4 Calculate the Period of the Oscillations To calculate the period of the spring, we will use the following equation: s 2π M + 1 m T = = 2π 3 spring (8) ω k

Here, M is the mass of the rubber bob and mspring is the mass of the spring. Using your numbers from the previous sections, calculate the theoretical period of the spring:

T = s

Calculate the percent difference between the theoretical and experimental value.

% Difference = %

3 Procedure: Simple Pendulum

A simple pendulum is a mass at the end of a very light string. We can treat the mass as a single particle and ignore the mass of the string, which makes calculating the rotational inertia very easy. A simple pendulum is expected to swing with a period such that: s L T = 2π (9) g Where L is the distance between the center of the mass to the axis of rotation. In this section, tie a mass to a string and attach it the holder. Then one partner should hold the holder to remove the effect of extra vibrations from the pendulum. When you are taking data, only pull the string to one side by approximately 10◦ or less for optimal results.

3.1 Measure the Period of the Oscillations Fill out the table below by measuring the period of oscillations for 10 oscillations. Repeat this process 3 times. Data Table Trial Time for 10 periods (seconds) Time for 1 period (seconds) 1

2

3

AVERAGE

3 3.2 Calculations for Simple Pendulum Calculate the theoretical period of your pendulum.

Length of Pendulum = m

Mass = m

T = s

Calculate the percent difference between the theoretical and experimental value.

% Difference = %

4 Rigid Pendulum

A physical pendulum is expected to swing with a period such that: s I T = 2π (10) mgh where m is the mass of the pendulum, I is the rotational inertia of the pendulum (including the use of the parallel axis theorem as needed), and h is the distance from the pendulum’s center of mass to the axis of rotation. For 2 this metal meter stick, rotating about a point 10 cm from the end of the stick) is Irot = 0.11 kg ∗ m . Measure the time it takes for 10 oscillations when the physical pendulum is rotating around the 10cm mark and calculate the expected period.

mass of pendulum = kg

Theoretical Period = s

Data Table Trial Time for 10 periods (seconds) Time for 1 period (seconds) 1

2

3

AVERAGE

% Difference = %

4 5 Problem Solving: The Simple Harmonic Motion of a Spring

A 2.3 kg mass oscillates back and forth from the end of a spring of spring constant 120 N/m. At t = 0, the position of the block is x = 0.13 m and its velocity is vx = -3.4 m/s.

1. What is the Angular frequency of the block?

2. What is the mechanical energy of this block-spring system?

3. What is the amplitude of the oscillation?

4. What is the maximum speed of the block and where is this experienced over the motion?

5. What is the phase constant? (Choose a cosine function to describe the motion)

6. What is the position of the block as a function of time?

7. What is the maximum acceleration of the block and where is this experienced over the motion?

5 6 Problem Solving: The Simple Harmonic Motion of a Pendulums

1. Working from the general experession for the something that oscillates sinusoidally and the particular result for q k ω for the block-spring oscillator: x(t) = xmaxcos(ωt + φ), ω = m show that the mechanical energy is a constant 1 2 Emech = 2 kxmax for all time.

2. By carrying out a Newton’s 2nd law for rotational analysis for a simple pendulum (a point mass on the end of a string), derive an expression for the period for small amplitudes. (We’ll be making use of the small angle approximation sin(θ) ≈ θ which is valid when θ is small and expressed in radians.)

3. A physical pendulum is a rigid extended object that is mounted to a pivot. Its essential features are its rotational inertia I about the pivot, its mass M, and the distance between the pivot and the center of mass hcom. A. Derive the general expression for its period for small amplitudes.

B. Calculate the period for a meter stick with the pivot mounted at one end.

C. Calculate the period for a meter stick with the pivot mounted at the 20 cm mark.

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