Sajjad Ahmed Memon S.S./ Health Physicist NIMRA 1 Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Simple Harmonic Motion S.H.M. Sajjad Ahmed Memon Senior Scientist NIMRA
Simple Harmonic Motion It is a type of periodic motion where the restoring force is directly proportional to the displacement. It can serve as a mathematical model of a variety of motions, such as the oscillation of a spring, motion of a simple pendulum etc. Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Mass attached with Spring Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Back and forth motion that is caused by a force that is directly proportional to the displacement. The displacement centers around an equilibrium position. Which is called Hooke’s Law Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring. Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA From Newton’s second law of motion Compare Newton’s second law of motion and Hooke’s Law simultaneously Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA The unit of “a” is meter/sec2 The unit of “m” is kg “x’” is measured in meters The unit of “k” is N/meter or kg/sec2 Where N=kg-meter/sec2 Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Also acceleration of a body in circular motion is given by Therefore Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Since a cycle is 2p radians, the relationship between frequency and angular velocity is: Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Therefore “f” is measured in sec-1 or Hz As “T” is in sec Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Simple Pendulum Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Forces acting on Bob/Ball 1. Weight of the bob (W) acting vertically downward. 2. Tension in the string (T) acting along the string. The weight of the bob can be resolved into two rectangular components: a. Wcosθ along the string. b. Wsinθ perpendicular to string. Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Since there is no motion along the string, therefore, the component Wcosθ must balance tension (T) i.e. Wcosθ = T This shows that only Wsinθ is the net force which is responsible for the acceleration in the bob of pendulum. According to Newton's second law of motion Wsinθ will be equal to ma Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA i.e. Wsin θ = ma Since Wsinθ is towards the mean position, therefore, it must have a negative sign. i.e. ma = - Wsinθ But W = mg ma = - mgsinθ a = - gsinθ In our assumption θ is very small because displacement is small, in this condition we can take sinθ = θ Hence a = - gθ Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA If x be the linear displacement of the bob from its mean position, then from figure, the length of arc is nearly equal to x From elementary geometry we know that: Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA The unit of “a” is meter/sec2 The unit of “l” is meter “x’” is measured in meters The unit of “g” is meter/sec2 Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Also acceleration of a body in circular motion is given by Therefore Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Since a cycle is 2p radians, the relationship between frequency and angular velocity is: Sajjad Ahmed Memon S.S./ Health Physicist NIMRA
Sajjad Ahmed Memon S.S./ Health Physicist NIMRA Therefore “f” is measured in sec-1 or Hz As “T” is in sec Sajjad Ahmed Memon S.S./ Health Physicist NIMRA