Chapter 11 Elasticity And Periodic Motion. Stress characterizes the strength of the force associated with the stretch, squeeze, or twist, usually on a “force per unit area” basis. Strain describes the deformation that occurs.
Elasticity And Periodic Motion
Stress characterizes the strength of the force associated with the stretch, squeeze, or twist, usually on a “force per unit area” basis.
Strain describes the deformation that occurs.
When the stress and strain are small enough, we often find that the two are directly proportional. The general pattern that emerges can be formulated as
Stress/Strain = Constant.
Experiments have shown that, for a sufficiently small tensile or compressive stress, stress and strain are proportional, as stated by Hooke’s law. The corresponding proportionality constant, called Young’s modulus (denoted by Y ), is given by
Y = (Tensile stress)/(Tensile strain) or
Y = (Compressive stress)/(Compressive strain),
Y = (F /A) / (Δl / l0)
Fperpendicular = (YAΔl) / l0
Let k = (YA)/ l0 , Δl = x and Fperpendicular = Fx ; then we have Fx = kx.
The pressure in a fluid, denoted by p, is the force Fperpendicular per unit area A transmitted across any cross section of the fluid, against a wall of its container, or against a surface of an immersed object:
p = Fperpendicular/A
When a solid object is immersed in a fluid and both are at rest, the forces that the fluid exerts on the surface of the object are always perpendicular to the surface at each point.
When Hooke’s law is obeyed, the volume strain is proportional to the volume stress (change in pressure). The corresponding constant ratio of stress to strain is called the bulk modulus, denoted by B.
When the pressure on an object changes by a small amount Δp, from p0 to p0 + Δp, and the resulting volume strain is ΔV/V, Hooke’s law takes the form
B = - (Δp)/(ΔV/V0)
We include a minus sign in this equation because an increase in pressure always causes a decrease in volume. In other words, when Δp is positive, ΔV is negative.
If the forces are small enough so that Hooke’s law is obeyed, the shear strain is proportional to the shear stress. The corresponding proportionality constant (ratio of shear stress to shear strain), is called the shear modulus, denoted by S:
S = Shear stress / Shear strain
= (Fparallel / A) / (x / h)
= (Fparallel / A) / Φ
Using Newton’s second law,
max = -kx
ax = -(k/m)x
Conservation of Mechanical Energy
E = (1/2) mvx2 + (1/2)kx2 = constant
When x = ± A, vx = 0. At this point, the energy is entirely potential energy and E = (1/2)kA2 .
E = (1/2)kA2 = (1/2) mvx2 + (1/2)kx2
vx = ± k/mA2 – x2
We can use this equation to find the magnitude of the velocity for any given position x.
The relationship between uniform circular motion and simple harmonic motion.
x = A cos θ
x = A cos(ωt)
x = A cos [(2π/T )t]
SI unit: m
v = -Aω sin(ωt)
SI unit: m/s
Maximum speed of the mass is
vmax = Aω
From equation ax = -Aω2 cos(ωt)
ax = -Aω2 (maximum acceleration) (1)
Also, we know that
ax = -(kx)/m
ax = -(kA)/m (maximum acceleration)(2)
From equations (1) and (2)
-Aω2 = -kx/m
ω2 = k/m
ω= k/m = 2πf = 2π/T
T = 2πm / k
SI unit: s
The restoring force F at each point is the component of force tangent to the circular path at that point:
F = -mgsinθ
If the angle is small, sinθis very nearly equal toθ(in radians).
F = -mgθ = -mgx/L
F = -(mg/L)x
The restoring force F is then proportional to the coordinate x for small displacements, and the constant mg/L represents the force constant k.
on the mass but on the length of the pendulum
A 0.85-kg mass attached to a vertical spring of force constant 150 N/m oscillates with a maximum speed of 0.35 m / s. Find the following quantities related to the motion of the mass: (a) the period, (b) the amplitude, (c) the maximum magnitude of the acceleration.
A peg on a turntable moves with a constant linear speed of 0.67 m / s in a circle of radius 0.45 m. The peg casts a shadow on a wall. Find the following quantities related to the motion of the shadow: (a) the period, (b) the amplitude, (c) the maximum speed, and the maximum magnitude of acceleration.
(d) If you do not change the pendulum’s length in part (c ), what is its period on that planet in terms of T?
(e) If a pendulum has a period T and you triple the mass of its bob, what happens to the period (in terms of T)?
Find the period, frequency, and angular frequency of (a) the second hand and (b) the minute hand of a wall clock.
CHAPTER 11, Problem 47
A certain simple pendulum has a period on earth of 1.60 s. What is the period on the surface on Mars, where the acceleration due to gravity is 3.71 m/s2 ?
An object suspended from a spring vibrates with simple harmonic motion. At an instant when the displacement of the object is equal to one-half the amplitude, what fraction of the total energy of the system is kinetic and what fraction is potential?
After you pick up a spare, your bowling ball rolls without slipping back toward the ball rack with a linear speed of 2.85 m/s. To reach the rack, the ball rolls up a ramp that rises through a vertical distance of 0.53 m. (a) What is the linear speed of the ball when it reaches the top of the ramp? (b) If the radius of the ball were increased, would the speed found in part (a) increase, decrease, or stay the same? Explain.
MECHANICAL WAVES AND SOUND
A disturbance that propagates from one place to another is referred to as a wave.
Waves propagate with well-defined speeds determined by the properties of the material through which they travel.
Waves carry energy.
In a transverse wave individual particles move at right angles to the direction of wave propagation.
In a longitudinal wave individual particles move in the same direction as the wave propagation.
As a wave on a string moves horizontally, all points on the string vibrate in the vertical direction.
The water wave has characteristics of both transverse and longitudinal waves.
Speed of Sound in Air
v = 343 m/s
The frequency of sound determines its pitch. High-pitched sounds have high frequencies; low-pitched sounds have low frequencies.
Human hearing extends from 20 Hz to 20, 000 Hz. Sounds with frequencies above this range are referred to as ultrasonic, while those with frequencies lower than 20 Hz are classified as infrasonic.
In phase/opposite phase: Two sources are in phase if they both emit crests at the same time. Sources have opposite phase if one emits a crest at the same time other emits a trough.
Constructive interference occurs when the path length from the two sources differs by 0, λ, 2λ, 3λ, …….
Destructive interference occurs when the path length from the two sources differs by λ/2, 3λ/2, 5λ/2, …….
The loudness of a sound is determined by its intensity; that is; by the amount of energy that passes through a given area in a given time.
I = E/(At)
I = P/A
I2 = P/(4πr22)