Hooke’s law for extended spring, F = -kl (a) (b) (c)
K= spring constant, l =extension, l0 = length of the spring [Fig (a)]
From Fig (b), At equilibrium condition, σ 𝐹 = 0
= mg – kl = 0, so, mg=kl
The Fig (c), The upward force the spring exerts on the body is k (l+y)
The downward force acting on the body is mg.
So, The resultant force on the body, σ 𝐹 = mg - k(l+y)= mg-kl-ky = -ky 𝑑2𝑦 Newton’s 2nd law of motion gives, σ 𝐹 = m 2 𝑑𝑡 𝑑2𝑦 Finally, m 𝑑𝑡2 = -ky Fig: A vertical spring-mass system. 𝑑2𝑦 𝑘 + 𝑚 y = 0; Same as the Diff. equation of SHM. 𝑑𝑡2
𝑑2𝑦 𝑘 + ω2y = 0 [ω = ] 𝑑𝑡2 𝑚 Difference between the horizontal and vertical spring-mass system
Horizontal and vertical mass-spring systems are both in simple harmonic motion.
• A vertical mass-spring system oscillates around the point where the downward force of gravity and the upward spring force cancel one another out (equilibrium position). The force of gravity only served to shift the equilibrium location of the mass. • The restoring force for a horizontal mass-spring system is just the spring force, because that is the net force in the x- direction. • The restoring force for a vertical mass-spring system is the net force in the y-direction which equals the spring force minus the force of gravity.
❑ How does the period of motion of a vertical spring-mass system compare to the period of a horizontal system (assuming the mass and spring constant are the same)?
Answer: The periods of a vertical and horizontal spring-mass system are the same as long as the mass and spring constant are identical. 𝒎 Time period, T = 𝟐𝝅 𝒌 Pendulums Pendulums are simple harmonic oscillators in which the springiness is associated with the gravitational force rather than with the elastic properties of a twisted wire or a compressed or stretched spring.
The Simple Pendulum
The tangential component Fgsin 𝜃 produces a restoring torque about the pendulum’s pivot point. we can write this restoring torque as 𝜏 = − L(Fg sin 𝜃) where the minus sign indicates that the torque acts to reduce 𝜃 and L is the moment arm of the force component Fgsin𝜃 about the pivot point.
I𝛼 = −L(mg sin 𝜃) [𝜏 = I𝛼 ] mgL 𝛼=− 𝐼 𝜃 [ sin 𝜃 ~ 𝜃, 𝑤ℎ𝑒𝑛 𝜃 𝑖𝑠 𝑠𝑚𝑎𝑙𝑙] 𝑚𝑔𝐿 𝛼 = − ω2 𝜃 [ω = 𝐼 ] The angular acceleration of a pendulum is proportional to its angular displacement but acts in the opposite direction, indicating simple harmonic motion. 𝑰 So, Time period, T = 𝟐𝝅 𝒎𝒈𝑳 𝑚𝐿2 T = 2𝜋 [ I = mL2 for the rotational inertia of the pendulum] 𝑚𝑔𝐿 𝐿 T = 2𝜋 𝑔 Why is the time period of a simple pendulum independent of the mass, while the time period of a spring-mass system depends on the mass?
Simple Pendulum: Time period does not depend on mass because the gravitational force and inertia cancel out.
Spring-Mass System: Time period depends on mass because inertia affects the system’s ability to oscillate
Physical Pendulum The physical pendulum has a moment arm of distance h about the pivot point
𝑰 Time period, T = 𝟐𝝅 [ now I is not simply mL2 ] 𝒎𝒈𝒉 What is the period of a physical pendulum if it is suspended at its center of mass?
A physical pendulum will not swing if it pivots at its center of
mass. This corresponds to putting h= 0 in. That equation then predicts T is infinity, which implies that such a pendulum will never complete one swing. When the pivot point is located exactly at the center of gravity, the r = 0, torque is zero. An angular version of a simple harmonic oscillator Differential equation: Torsional pendulum Hooke’s law for angular motion, • 𝜏 = −κ𝜃 κ = torsional spring constant [depends on the length, diameter, and material of the suspension wire] Newton’s 2nd law for angular motion, 𝑑2𝜃 𝜏 = 𝐼𝛼 = 𝐼 2 𝑑𝑡 𝑑2 𝜃 Equating expressions, −κ𝜃 = 𝐼 𝑑𝑡2 𝑑2 𝜃 κ 𝑑2 𝜃 + 𝜃= + ω2𝜃=0 𝑑𝑡2 𝐼 𝑑𝑡2
Solution of the diff. equation, 𝜃 = 𝜃𝑚𝑠𝑖𝑛 𝜔𝑡 + 𝜑
Fig: A torsional pendulum consists of a rigid body suspended by a 𝜃 = angular displacement wire. The rigid body oscillates between θ = + θ and θ = -θ. κ 𝜃𝑚= angular amplitude, ω = angular frequency = 𝐼
Significance 𝐼 Time period, T = 2π κ Like the force constant of the system of a spring – mass, the larger the torsion constant, the shorter the period. ➢ Show that the unit for the period is the second. ❑ A rod has a length of l = 0.30 m and a mass of 4.00 kg. A string is attached to the CM of the rod and the system is hung from the ceiling. The rod is displaced 10 degrees from the equilibrium position and released from rest. The rod oscillates with a period of 0.5 s. What is the torsion constant κ ? Ans: 4.73 N · m.
❑ Fig shows a thin rod whose length L is 12.4 cm and whose mass m is 135 g, suspended at its midpoint from a long wire. Its period Ta of angular SHM is measured to be 2.53 s. An irregularly shaped object, which we call object X, is then hung from the same wire, as in Fig, and its period Tb is found to be 4.76 s. What is the rotational inertia of object X about its suspension axis ? Ans: 6.12 ×104 kg m2.
❑ The balance wheel of an old-fashioned watch oscillates with angular amplitude π rad and period 0.500 s. Find (a) the maximum angular speed of the wheel, (b) the angular speed at displacement π/2 rad, and (c) the magnitude of the angular acceleration at displacement π/4 rad. Check Point! ❑ Two simple pendulums are constructed by an engineer, each hanging from a wire attached to the ceiling. Both pendulums are positioned 2 cm above the floor. The first pendulum has a bob with a mass of 10 kg, while the second pendulum's bob has a mass of 100 kg. Describe how the motion of the pendulums will differ if the bobs are both displaced by 12˚.
➢ Their motions will be similar in terms of the period and frequency of their swings, as the period of a simple pendulum is independent of the mass of the bob.
❑ If frequency is not constant for some oscillation, can the oscillation be SHM? (b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?
❑ Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.
❑ Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a more pliable material.
