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Simple Harmonic Motion and Uniform Circular Motion are two fundamental concepts in physics that play a crucial role in understanding the motion of objects in various systems.
- Simple Harmonic Motion (SHM) deals with the repetitive back-and-forth motion of an object around a stable equilibrium position, exhibiting sinusoidal behaviour.
- It is encountered in various natural phenomena, such as a swinging pendulum, vibrating strings, and oscillating springs.
- Uniform Circular Motion (UCM) involves the motion of an object moving at a constant speed in a circular path.
- The study of UCM is vital in comprehending the dynamics of celestial bodies, rotating machinery, and many real-world applications.
Key Terms: Simple Harmonic Motion, Circular Motion, Modulus of Elasticity, Uniform Circular Motion, Motion Projection, Spring
What is Simple Harmonic Motion (SHM)?
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A stone attached to the end of a string would travel uniformly in a circle around a fixed point if it is moved in a horizontal plane at a constant angular speed. The stone will appear to move to and fro along the horizontal line with the other end of the string serving as the halfway point if it is looked sideways.
- Similar to this, the stone's projected motion or shadow would appear to move back and forth perpendicular to the circle's arc.
- Galileo noticed a similar phenomenon when he realized Jupiter's four main moons moved back and forth about the planet in a straightforward harmonic motion.
Simple Harmonic Motion (SHM)
- A particle P moving in a uniform circular motion with a radius of A and a constant angular speed ω is depicted in the above figure.
- It can be seen that the particle's location at every time t is ωt + φ where φ is the particle's angular position in the specified circle at any time t.
- A different particle, Q is used to define the projection of the particle on the X-axis.
- The particle P's projection on the X-axis can be expressed as, x(t) = A cos (ωt + φ)
- The position vector of particle Q, which is moving in a simple harmonic manner, is provided by the expression above.
- Particle Q is the projection of particle P.
- The expression provides the particle Q's location vector at any time t.
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| Angular Velocity | Period Angular Frequency | Oscillatory Motion |
Solving Problems on SHM (Simple Harmonic Motion)
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Assume two particles A and B are operating according to the straightforward harmonic equation shown below:
XA = A sin ( ωt ) XB = A sin ( ωt + π3 )
- The interval after which these two particles will initially collide must be determined.
- When B is in the fourth quadrant and A is in the first quadrant, it can be claimed that both projections of A and B will meet somewhere.
- However, the angle between them will remain the same because both are revolving at the same angular speed.
- So let's say they cross paths after A has altered his position θ. As shown in the following figure:
As A and B have the same projection on the x-axis
so, ∠AOX = ∠BOX …….(1)
And it is known, ∠AOB = π3 …….(2)
From (1) and (2) it can be said that, ∠AOX = π6
This leads to the conclusion that A's angle of motion is π3 and its angular speed is ω.
Therefore, time can be calculated as follows: t = π3ω s
- In a similar vein, it can be argued that the oscillation's period is 2πω s.
- If the aforementioned scenario is considered, the magnitude of velocity will be the same but the direction of movement cannot be equal.
- Therefore, based on analysis of this situation, it may be predicted that B will be located somewhere in the third quadrant and A in the fourth.
- Additionally, the extent of their deviation from the mean position will be the same.
Given that the displacement should have a similar magnitude
so ∠AOY = ∠BOY ……..(3)
And it is known that, ∠AOB = π3 …….(4)
From (3) and (4) it can be said that, ∠AOY = π 6
So total angle covered by A = π - π 6
7π 6
Time taken = 5π6ω s
If a particle undergoes uniform circular motion, its projection can be described as exhibiting simple harmonic motion, where the diameter of the circle serves as the axis of oscillation. In other words, simple harmonic motion can be visualized as the projection of uniform circular motion onto the diameter of the circle where the circular motion is taking place.
SHM Circular Motion
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A displacement is created when a reference particle moves around a circle of reference.
Where the term "amplitude," indicated by the letter "A," refers to a particle's maximum departure from its location. It is equivalent to a circle's radius.
If the span of S.H.M. is S, then:
Amplitude A = S/2
Simple Harmonic Motion (SHM) in a Uniform Circular Motion
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From (1),
y = a Sinωt
Differentiating both sides
d(y)/dt = a d(Sinωt)dt
V = a ω Cosωt = a ω √(1 - Sin2ωt)
= ω√ (a2 – y2)
At mean position, y = 0, then
V = a ω
At the extreme position, y = a
V = 0
As a result, the velocity amplitude is the highest velocity in SHM for a body in a uniform circular motion.
Modulus of Elasticity
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A measurement of an object's stress-strain connection is its modulus of elasticity. The most crucial element in determining how much concrete will bend in response to applied stress is its elastic modulus.
The constants known as elastic constants control the deformation that results from a certain stress system acting on a material.
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Projecting Uniform Circular Motion and SHM (Simple Harmonic Motion) of a Spring
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To the ball and spring set, attach a ring stand. Place the vertical-plane spinning motor next to the spring. When the illuminated slide projector is held at a distance of about a meter, images of the ball rotating and oscillating must be displayed on the spring and the wall, respectively. Set the mass's oscillation to begin in a phase that is equal to that of the driven ball.
Uniform circular motion is the description of a substance moving at a constant speed along a circular route. The one-dimensional projection of this motion can be modeled as a straightforward harmonic motion.
Things to Remember
- Simple Harmonic Motion (SHM) is a back-and-forth motion with equal displacements on both sides of equilibrium, repeating after a set time.
- By projecting uniform circular motion onto the circle's diameter where the circular motion occurs, one can visualize the nature of Simple Harmonic Motion (SHM).
- SHM equation: x(t) = A cos(ωt + φ), where A is amplitude, ω is angular speed, t is time, and φ is phase angle.
- The period (T) and frequency (f) of SHM are related to the angular frequency (ω) by T = 2π/ω and f = 1/T.
- Modulus of Elasticity measures an object's stress-strain relationship, determining its response to stress.
- Uniform circular motion is constant-speed motion along a circular path.
- Periodic motion refers to a body's motion that repeats itself after a fixed interval of time.
- SHM represents 1D circular motion with uniform circular motion's angular speed ω and x-axis projection undergoing SHM.
Previous Year Questions
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Sample Questions
Ques: What are periodic motion and circular motion? (1 Mark)
Ans: A body that forms a circular locus can do so partially or completely, repeating itself after a predetermined amount of time. If such a circular motion repeats itself after each 360° rotation, it may qualify as a periodic motion.
Ques: Is circular motion Simple Harmonic Motion? (1 Mark)
Ans: Yes, basic harmonic motion can be used to represent one-dimensional circular motion. A particle P moves uniformly in a circular motion at a constant speed ω along the circle of reference. Its x-axis projection experiences a straightforward harmonic motion.
Ques: Are circular motion and rotation equivalent? (2 Marks)
Ans: The following lists the distinctions between rotational motion and circular motion:
Circular motion stands as a unique illustration of rotational motion, characterized by a constant distance between the body's center of mass and the axis of rotation. Rotational motion is the motion of a body around its center of mass.
Ques: Give examples of circular motion. (2 Marks)
Ans: The following are some instances of circular motion in real life:
- The motion of a ball on a rope spinning
- Planets' rotation around the sun.
- Navigating a curve on a racecourse with an automobile.
Ques: What does the term Simple harmonic motion (SHM) mean? (2 Marks)
Ans: Simple harmonic motion is a type of repeated back-and-forth motion in physics where the maximum displacement on one side of the equilibrium position is equal to the maximum displacement on the opposite side. The same time interval characterizes each entire vibration.
It is a very important subject with considerable applications and connections to other important physics subjects. To have a comprehensive understanding of the procedures involved in any SHM scenario, numerical SHM must also be practiced.
Ques: How are the period and frequency of a simple harmonic motion (SHM) calculated? (1 Mark)
Ans: Period is given by T = 2π/ω and Frequency = 1/T and angular frequency ω = 2πf = 2π/T.
Ques. Are circular motion and rotation equivalent? (3 Marks)
Ans. No, circular motion and rotation are not equivalent. While circular motion refers to the movement of an object along a circular path at a constant speed, rotation refers to the motion of a body around its center of mass.
- In a circular motion, the distance between the object's center and the axis of rotation remains constant, whereas, in rotational motion, the object spins around an internal axis.
- However, it is important to note that circular motion is a specific example of rotational motion, where the axis of rotation lies outside the object, and the object moves along a circular path.
- On the other hand, general rotational motion involves spinning around the object's center of mass, and its axis may not necessarily coincide with the circular path.
Ques. What is the period of Simple Harmonic Motion (SHM)? (2 Marks)
Ans. The period of Simple Harmonic Motion (SHM) is the time taken for one complete oscillation or cycle. It represents the time interval required for the object to return to its initial position after completing one back-and-forth motion. The period (T) of a system is mathematically related to the angular frequency (ω) through the equation T = 2π/ω.
It is an important parameter in SHM, as it determines the frequency (f) of the oscillation, where f = 1/T. The period is independent of the amplitude of oscillation and remains constant for a given system undergoing SHM.
Ques. How does the amplitude affect Simple Harmonic Motion (SHM)? (3 Marks)
Ans. The amplitude (A) in Simple Harmonic Motion (SHM) represents the maximum displacement of the object from its equilibrium position.
- The amplitude directly influences the extent of the oscillation, determining how far the object moves from its resting position during each cycle.
- A larger amplitude results in a more significant displacement and wider oscillation, while a smaller amplitude leads to a lesser displacement and narrower oscillation.
- However, it is important to note that the amplitude does not affect the period or frequency of SHM, as these parameters remain constant for a given system regardless of the amplitude.
Ques. What is the restoring force in Simple Harmonic Motion (SHM)? (3 Marks)
Ans. In Simple Harmonic Motion (SHM), the restoring force is the force that acts on the object to bring it back to its equilibrium position when displaced from it. The magnitude of the restoring force in SHM is directly related to the displacement from the equilibrium position and opposes the direction of the displacement.
For example, if the object is displaced to the right, the restoring force will act to the left, attempting to bring the object back to its resting position. This relationship is described by Hooke's Law, which states that the magnitude of the restoring force (F) is given by F = -kx, where k is the spring constant and the displacement from the equilibrium position is denoted by x.
Ques. How is Simple Harmonic Motion (SHM) related to energy? (3 Marks)
Ans. In Simple Harmonic Motion (SHM), the total mechanical energy of the oscillating system remains constant throughout the motion. As the object oscillates back and forth, it interconverts between kinetic energy (KE) and potential energy (PE).
- At the equilibrium position, where the displacement is zero, the object's velocity is maximum, and it possesses only kinetic energy.
- As the object moves away from the equilibrium position, its kinetic energy decreases while its potential energy increases, reaching a maximum at the amplitude.
- At the amplitude, the kinetic energy is zero, and the object has only potential energy.
- The total mechanical energy (TE) in SHM stays constant as it is the sum of kinetic and potential energy.
Ques. What are the factors affecting the frequency of Simple Harmonic Motion (SHM)? (3 Marks)
Ans. The frequency (f) of Simple Harmonic Motion (SHM) represents the number of complete oscillations or cycles per unit of time. The frequency is inversely proportional to the period (T) and is given by f = 1/T. Therefore, the factors affecting the frequency are the same as those affecting the period.
The frequency is determined by the angular frequency (ω), which, in turn, is related to the mass (m) and the spring constant (k) of the system. For a mass-spring system, the frequency is given by f = 1/(2π) * √(k/m). The frequency increases with an increase in the spring constant or a decrease in the mass, indicating a faster oscillation rate.
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