Principles of Mechanics pp 155-171 | Cite as

# Oscillatory Motion

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## Abstract

A motion repeating itself is referred to as periodic or oscillatory motion. An object in such motion oscillates about an equilibrium position due to a restoring force or torque. Such force or torque tends to restore (return) the system toward its equilibrium position no matter in which direction the system is displaced. This motion is important to study many phenomena including electromagnetic waves, alternating current circuits, and molecules. For a vibration to occur, two quantities are necessary to be present—stiffness and inertia.

## 10.1 Oscillatory Motion

A motion repeating itself is referred to as periodic or oscillatory motion. An object in such motion oscillates about an equilibrium position due to a restoring force or torque. Such force or torque tends to restore (return) the system toward its equilibrium position no matter in which direction the system is displaced. This motion is important to study many phenomena including electromagnetic waves, alternating current circuits, and molecules. For a vibration to occur, two quantities are necessary to be present—stiffness and inertia.

## 10.2 Free Vibrations

When a system vibrates, a restoring force must be present. In addition to that force, there is always a retarding or damping force such as friction. If the effect of the damping force is small and can be neglected, then the motion is classified as free and undamped motion. Otherwise, the motion is classified as free damped motion. In both cases, the motion is known as free vibration since no forces other than the restoring and damping forces exist during vibration. If a driving force that does positive work on the system exists, the motion is classified as forced vibration.

This force may be applied externally to the system or sometimes is produced within the system. In this chapter, the case in which a restoring force is directly proportional to the displacement is considered. The resulting motion is then known as a harmonic vibration and the system is said to be linear. If the restoring force depends on the displacement in some other way, the resulting motion is known as anharmonic vibration and the system is said to be nonlinear.

## 10.3 Free Undamped Vibrations

This kind of motion is known as the simple harmonic motion. Next, we will examine examples of such motion in physics.

### 10.3.1 Mass Attached to a Spring

*m*attached to a light spring of spring constant

*k*that is fixed at the other end (see Fig. 10.1). Suppose that the system lies on a frictionless horizontal surface. For small displacements, the restoring force acting on the block by the spring is given by Hook’s law

*A*and \(-A\). The quantity \((\omega _{n}t-\phi )\) is called the phase angle. If this angle is increased by \( 2\pi \), all physical quantities such as the displacement, velocity, and acceleration repeat themselves. The plot of

*x*versus

*t*is shown in Fig. 10.2. If

*A*is fixed and \(\phi \) is changed the motion will be the same except that the same physical quantities will appear either earlier or later than the preceding motion.

#### 10.3.1.1 The Period and Frequency of Motion

#### 10.3.1.2 The Phase Difference

#### 10.3.1.3 The Velocity and Acceleration

#### 10.3.1.4 Boundary Conditions

*A*and \(\phi \) for a specific vibration. Suppose that the vibration is measured when the stopwatch is set to zero, i.e., at \(t=0\) and that at that instant the mass is released from rest at a distance of \(x=A_{1}\) from its equilibrium position. Substituting these conditions into Eqs. 10.3 and 10.4, we have

### Example 10.1

An object oscillates in simple harmonic motion according to the expression \(x=(3\mathrm {m})\cos (\pi t+\pi /3)\). Find (a) the amplitude, phase constant, period, and frequency of motion; (b) the displacement, velocity, and acceleration of the object at \(t=0.5\mathrm {s}(\mathrm {c})\) the time when the object first reach \(x=-1.5 \; \mathrm {m}.\)

### Solution 10.1

### Example 10.2

A 9 kg object is moving along the \(\mathrm {x}\)-axis under the influence of a force given by \(F=(-3x)\) N. Find (a) the equation of motion; (b) the displacement of the mass at any time if at \(t=0, x=5 \; \mathrm {m}\) and \(v=0.\)

### Solution 10.2

### Example 10.3

A 0.3 kg block is attached to a spring of force constant 20 \(\mathrm {N}/\mathrm {m}\) on a frictionless horizontal surface. If the initial displacement and velocity of the system is 0.02 \(\mathrm {m}\) and 0.2 \(\mathrm {m}/\mathrm {s}\), respectively, find the period, amplitude, and phase constant of motion.

### Solution 10.3

### Example 10.4

A particle of mass *m* is dropped in a straight tunnel that is drilled through the earth and which passes through the center of earth as shown in Fig. 10.5. Show that the motion of the particle is simple harmonic motion and find its period.

### Solution 10.4

### Example 10.5

A 0.4 kg block is connected to two springs of force constants \(k_{1}=20 \; \mathrm {N}/\mathrm {m}\) and \(k_{2}=50 \; \mathrm {N}/\mathrm {m}\) as in Fig. 10.6. Find (a) the total force acting on the block; (b) the period of motion.

### Solution 10.5

### Example 10.6

A 6 kg block is connected to a light spring of force constant of 300 \(\mathrm {N}/\mathrm {m}\) on a frictionless horizontal surface. On top of it a second block of mass of 2 kg is placed. If the coefficient of static friction between the two blocks is 0.4 (see Fig. 10.7), find the maximum amplitude the system can have when it is in simple harmonic motion such that there is no slipping between the blocks.

### Solution 10.6

### 10.3.2 Simple Harmonic Motion and Uniform Circular Motion

*A*centered at the \(\mathrm {x}\) and \(\mathrm {y}\) axes as shown in Fig. 10.8. Let A be the position vector of a particle \(\mathrm {P}\) rotating with a constant angular speed \(\omega _{n}\) in the anticlockwise direction. The particle is thus in uniform circular motion. Suppose \(\mathrm {P}\) starts the rotation at \(t=0\) at an angle of \(\phi \) measured from the positive \(\mathrm {x}\)-axis. At any time, the angular position of the particle is given by \((\omega _{n}t+\phi )\), therefore the vector position of the particle at any time is

### 10.3.3 Energy of a Simple Harmonic Oscillator

*U*and

*K*with time is at twice the angular frequency of the variation of

*x*,

*v*, and

*a*with time. This is because the potential energy is converted to kinetic energy twice in each cycle. The velocity of the simple harmonic oscillator can be obtained from the total energy of the system. From Eq. 10.10, we have

### Example 10.7

A 0.3 kg mass is attached to a light spring. If the total energy of the system is 0.025 \(\mathrm {J}\) and the amplitude of motion is 5 cm, find the period and frequency of motion.

### Solution 10.7

### Example 10.8

A 0.2 kg block is attached to a light spring of force constant of 11 \(\mathrm {N}/\mathrm {m}\) on a horizontal frictionless surface. If the block is displaced a distance of 8 cm from its equilibrium position, find (a) the amplitude, the angular frequency, the period and the frequency of motion when the block is released; (b) the maximum force exerted on the block; (c) the total mechanical energy of the system; (d) the maximum speed and maximum acceleration of the block; (e) the velocity of the block when its displacement is 2 cm; (f) the acceleration of the block when its displacement is 3 cm.

### Solution 10.8

### Example 10.9

An object connected to a spring is in simple harmonic motion on a frictionless surface. If the object’s displacement when \((2v_{\max }/3)\) is \(\pm 0.015 \; \mathrm {m}\), find the amplitude of motion.

### Solution 10.9

### Example 10.10

A solid cylinder is connected to a light spring as in Fig. 10.12. If the cylinder rolls without slipping along the surface, show that the motion of the cylinder is simple harmonic motion and find its frequency.

### Solution 10.10

### 10.3.4 The Simple Pendulum

*L*that is fixed at the other end (see Fig. 10.13). If the mass is pulled to the right or left from its equilibrium position and released, then the pendulum will swing in a vertical plane about an axis passing through O. The resulting motion is then a periodic or oscillatory motion. The restoring torque is due to gravity and is given by

#### 10.3.4.1 Energy

### Example 10.11

A simple pendulum is 0.5 \(\mathrm {m}\) long. Find its period at the surface of Mars and compare it to its period at the earth’s surface.

### Solution 10.11

### Example 10.12

A simple pendulum of length of 2 \(\mathrm {m}\) is displaced through an angle of \(12^{\circ }\) and released. Find (a) the angular frequency of motion; (b) the maximum angular speed and maximum angular acceleration.

### Solution 10.12

### Example 10.13

A simple pendulum 1.4 \(\mathrm {m}\) in length is displaced through an angle of \(10^{\circ }\) and released. Find the velocity of the bob when it reaches the bottom.

### Solution 10.13

### 10.3.5 The Physical Pendulum

*d*from the center of mass. The equilibrium position of the body is when its center of mass is directly below the pivot O. If the body is displaced either to the right or left from the equilibrium position, a restoring torque due to gravity will act on it. As a result, the body will oscillate in a vertical plane where the axis of rotation is perpendicular to the page. The restoring torque is given by

*M*is the mass of the body and

*d*is the moment arm of the tangential component of the weight \((Mg\ \sin \theta )\). From Newton’s second law, we have

*m*, the moment of inertia is

*d*represents the length of the string.

### Example 10.14

A uniform rod of length of 0.6 \(\mathrm {m}\) that is suspended at one end oscillates with a small amplitude as in Fig. 10.17. Find the frequency of motion.

### Solution 10.14

### Example 10.15

A uniform square plate of length *a* is pivoted at one of its corners and oscillates in a vertical plane as in Fig. 10.18. Find the period of motion if the amplitude is small.

### Solution 10.15

### 10.3.6 The Torsional Pendulum

*k*is called the torsional constant. Its value depends on the property of the wire. Note that this equation is the rotational analogue of Hook’s law in linear form \((F=-kx)\). From Newton’s second law, we have

### Example 10.16

A uniform solid sphere of mass of 4.7 kg and radius of 5 cm is suspended at its midpoint by a light string (see Fig. 10.20) where it oscillates as a torsional pendulum. If the period of motion is 3.5 \(\mathrm {s}\), find the torsion constant.

### Solution 10.16

## 10.4 Damped Free Vibrations

*b*is a positive constant called the damping coefficient. Its SI units is \(\mathrm {N}(\mathrm {m}\,\mathrm {s}^{-1})=\mathrm {k}\mathrm {g}\,\mathrm {s}^{-1}\). The negative sign shows that the direction of the force is always opposite to the velocity. Now consider the spring–mass system as shown in Fig. 10.21, the cylinder shown in the figure contains a viscous fluid and a piston moving in it. Such device is known as the viscous damper. The net force on the oscillating body is

### 10.4.1 Light Damping (Under-Damped) \((\gamma <2\omega _{n})\)

*A*is the initial amplitude of motion. \(Ae^{\frac{-\gamma }{2}t}\) is called the amplitude of motion and \(\phi \) is the phase constant and \(\omega _{D}\) is the angular frequency of the damped motion. This equation shows that the system oscillates in a decreasing harmonic motion where the amplitude of motion decreases exponentially with time until eventually the oscillation dies out (see Fig. 10.22). The dashed lines in Fig. 10.22 are called the envelope of the oscillation curve. The period of motion in light damping is therefore given by

### Example 10.17

An 8 kg block is attached to a light spring and a light viscous damper. If at \(t=0, x=0.12 \; \mathrm {m}\) and \(v=0\), find (a) the displacement at any time; (b) the logarithmic decrement. \((k=30 \; \mathrm {N}/\mathrm {m},\ b=20 \; \mathrm {N}\,\mathrm {s}/\mathrm {m})\).

### Solution 10.17

*A*and \(\phi \) gives \(\phi =-0.7\) rad and \(A=0.17 \; \mathrm {m}.\) Therefore,

### 10.4.2 Critically Damped Motion \((\gamma =2\omega _{n})\)

### 10.4.3 Over Damped Motion (Heavy Damping) \((\gamma >2\omega _{n})\)

*t*unlike critical damping (see Fig. 10.24).

### Example 10.18

In Example 10.17, find the range of values of the damping coefficient for the system to be: (a) over damped; (b) critically damped.

### Solution 10.18

(a) over damped if \(\gamma >2\omega _{n}\), i.e., if \(\gamma >3.8\mathrm {s}^{-1}(\mathrm {b})\) critically damped if \(\gamma =3.8\mathrm {s}^{-1}.\)

### 10.4.4 Energy Decay

## 10.5 Forced Vibrations

### Example 10.19

In Example 10.17, if a driving force of the form \(F(t)=5\cos 4t\) is applied to the system, find the steady-state displacement as a function of time.

### Solution 10.19

### Example 10.20

In Example (10.17), find the steady-state displacement as a function of time if there is no damping.

### Solution 10.20

**Problems**

- 1.
A 2 kg block is fastened to a spring of force constant 98 \(\mathrm {N}/\mathrm {m}\) on a horizontal frictionless surface. If the block is released a distance of 6 cm from its equilibrium position, find (a) the angular frequency, the frequency and the period of the resulting motion, (b) the time it takes the block to first reach \(x=-5\) cm and its velocity at that time, (c) the maximum speed and maximum acceleration of the oscillating block, (d) the total mechanical energy of the oscillator.

- 2.
A 10 kg block is attached to a light spring of force constant 200 \(\mathrm {N}/\mathrm {m}\) on a smooth horizontal surface. Find the amplitude of motion if at \(x=0.06 \; \mathrm {m}\) the velocity of the block is \(v=0.5 \; \mathrm {m}/\mathrm {s}.\)

- 3.
A particle rotate counterclockwise in a circle of radius 0.2 \(\mathrm {m}\) with a constant angular speed of 2 \(\mathrm {r}\mathrm {a}\mathrm {d}/\mathrm {s}\). If at \(t=0\) the \(\mathrm {x}\)-coordinate of the particle is 0.14 \(\mathrm {m}\), find the displacement, velocity and acceleration of the particle at any time.

- 4.
If a simple pendulum has a period of 2 \(\mathrm {s}\), find its period when its length is increased by \(20\%\).

- 5.
A simple pendulum of length lm and mass of 0.4 kg oscillates in a region where \(g=9.8 \; \mathrm {m}/\mathrm {s}^{2}\). If the amplitude of oscillation is \(10^{\circ }\), find (a) the angular displacement, angular velocity and angular acceleration of the pendulum as a function of time.

- 6.
A uniform solid cylinder of radius

*R*and mass*M*rolls without slipping on a track of radius 4*R*as shown in Fig. 10.26. Find the period of oscillation when the cylinder is displaced slightly from its equilibrium position. - 7.
A planer body of mass 3 kg oscillates as a physical pendulum. If the period of oscillation is 3 \(\mathrm {s}\) and if the pivot point is at 0.2 \(\mathrm {m}\) from the center of mass, find the moment of inertia of the body.

- 8.
A uniform hollow cylinder of radius

*R*and mass*M*is suspended at its midpoint from a wire and form a torsional pendulum. If the period of motion is*T*, find the torsion constant. - 9.
For the system shown in Fig. 10.27, determine the displacement of the block at any time if at \(t=0, x=0\) and \(v=0.\,(k=200 \; \mathrm {N}/\mathrm {m},\ b=200 \; \mathrm {N}\,\mathrm {s}/\mathrm {m})\).

- 10.
For the system shown in Fig. 10.28, find the steady-state displacement as a function of time.

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