In this assignment, I will learn about the outcome two that is Newton’s law and harmonic oscillation. Newton’s law can be divide by three types that is 1st law, 2nd law and 3rd law. It is teach about the motion in our real life. Thus, harmonic oscillation can be divided by three types that are pendulum oscillation, damped oscillation and mechanic oscillation. All of these oscillation are useful in our life especial is use in different type of mechanics.
Question One
Research on the Newton’s Laws of motion, and make a report that provide detail explanation and examples on Newton’s 3 laws of motion. You report should include relevant and useful formula.
Answer
Newton’s law of motion can be divided by three types that is 1st law, 2nd law and 3rd law and it is law of gravity. The three laws are simple and sensible.
The first law states that a force must be applied to an object in order to change its velocity. When the object’s velocity is changing that mean it is accelerating, which implies a relationship between force and acceleration.
The second law, the acceration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the acceleration is in the direction of the net force acting on the object.
Finally, the third laws, whenever we push on something, it pushes back with equal force in the opposite direction.
Forces
A force is commonly imagined as a push or a pull on some object, perhaps rapidly, as when we hit a tennis ball with a racket. (see figure 1.0). We can hit the ball at different speeds and direct it ionto different parts of the opponents;s court. This mean that we can control the magnitude of the applied force and alos its direction, so force is a vector quantity, just like velocity and acceleration.
Figure 1.0: Tennis champion Rafael Nadal strikes the ball with his racket, applying a force and directing the ball into the open part of the court.
Figure 1.1: Examples of forces applied to various objects. In each case, a force acts on the object surrounded by the dashed lines. Something in the environment external to the boxed area exerts the force.
Newton’s 1st law
Newton’s 1st law of motion states that if a body is at rest it will remain at the rest and if a body is moving in a straight line with uniform velocity will keep moving unless an external force is acted upon.
For example, consider a book lying on a table. Obviously, the book remains at rest if left alone. Now imagine pushing the book with a horizontal force great enough to overcome the force of friction between the book and the table, setting the book in motion. Because the magnitude of the applied force exceeds the magnitude of the friction force, the book to a stop.
Now imagine the book across a smooth floor. The book again comes to rest once the force is no longer applied, but not as quickly as before. Finally, if the book is moving on a horizontal frictionless surface, it continues to move in a straight line with constant velocity until it hits a wall or some other obstruction.
However, an object moving on a frictionless surface, it’s not the nature of an object to stop, once set in motion, but rather to continues in its original state of motion. This approach was later formalized as Newton’s first law of motion:
An object moves with a velocity that is constant in magnitude and direction, unless acted on by a nonzero net force.
For example:
In the figure 1.2, the string is providing centripetal force to move the ball in a circle around 3600. If sudden the string was break, the ball will move off in a straight line and the motion in the absence of the constraining force. This example is not have other net forces are acting, such as horizontal motion on a frictionless surface.
Figure 1.2
Inertia
Inertia is the reluctance of an object to change its state of motion. This means if an object is at rest it will remain at rest or if it’s moving it will keep moving in a straight line with uniform velocity. Force is needed to overcome inertia.
For example
In figure 1.3, it is an experiment to prove the concept of inertia. In experiments using a pair of inclined planes facing each other, Galileo observed that a ball would up the opposite plane to the same height and roll down one plane. If smooth surface are used, the ball is roll up to the opposite plane and return to the original height.
When it is starting to roll down the ball on the degree place, it is will return the ball at the same height from original point.
Figure 1.3
If the opposite incline were elevated at nearly a 0 degree angle, then the ball will be roll in an effort to reach the original height that is show in the figure 1.4.
Figure 1.4: If a ball stops when it attains its original height, then this ball would never stop. It would roll forever if friction were absent.
Other example
Figure 1.5: According to Newton’s 1st law, a bicycles motion wasn’t change until same force, such as braking makes it change.
Newton 2nd law
Newton’s first law explains what happens to an object that has no net force acting on it. The object either remains at rest or continues moving in a straight line with constant speed. Newton’s second law is the acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the acceleration is in the direction of the acceleration is in the direction of the net force acting on the object.
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Imagine pushing a block of ice across a frictionless horizontal surface. When you exert some horizontal force on the block, it moves with an acceleration of the 2m/s2. If you apply a force twice as large, the acceleration doubles to 4m/s2. Pushing three times as hard triples the acceleration, and so on. From such observations, we conclude that the acceleration of an object is directly proportional to the net force acting on it.
Mass also affects acceleration. Suppose you stack identical block of ice on top of each other while pushing the stack with constant force. If the force applied to one block produces an acceleration of 2m/s2, then the acceleration drops to half that value, 1 m/s2, When 2 blocks are pushed, to one-third the initial value. When three block is pushed, and so on. We conclude that the acceleration of an object is inversely proportional to its mass. These observations are summarized in Newton’s second law:
The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Units of Force and Mass
The SI unit of force is the Newton. When 1 Newton of force acts on an object that has a mass of 1 kg, it produces an acceleration of 1 m/s2 in the object. From this definition and Newton’s second law, we can see that the Newton can be expressed in terms of the fundamental units of mass, length and time.
1 N = 1 kg.m/s2
A force is a push or a pull. Hence a force can change the size, shape, and state of rest or motion, direction of motion and speed / velocity. The symbol for force is F and the S.I. unit is Newton (N). An object of mass m is subjected to a force F, its velocity changes from U to V in time t. The above condition can be stated as:
F =
Where a = is acceleration, thus F = ma.
For example
Figure 1.6: An airboat.
An airboat with mass 3.50x102Kg, including passengers, has an engine that produces a net horizontal force of 7.70x102N, after accounting for forces of resistance (see figure 1.6).
(a) Find the acceleration of the airboat.
(b) Starting from rest, how long does it take the airboat to reach a speed of 12.0m/s2?
(c) After reaching this speed, the pilot turns off the engine and drifts to a stops over distance of 50.0m. Find the resistance force, assuming it’s constant.
Solution
(a) Find the acceleration of the airboat.
Apply Newton’s second law and solve for the acceleration:
Fnet = ma
a = =
= 2.20m/s2
(b) Find the time necessary to reach a speed of 12.0m/s.
Apply the kinematics velocity equation:
If t = 5.45s
V = at + V0 = (2.20m/s2) (5.45) = 12.0m/s
(c) Find the resistance force after the engine is turned off.
Using kinematics, find the net acceleration due to resistance forces
V2 – = 2a Δx
0 – (12.0m/s)2 = 2a(50.0m)
= -12 / 100
= -0.12m/s2
Substitute the acceleration into Newton’s second law, finding the resistance force:
Fresistance= ma
= (3.50 X 102kg) (-144m/s2)
= -504N
Impulse and Impulsive Force
The force, which acts during a short moment during a collision, is called Impulsive Force. Impulse is defined as the change of momentum, so Impulse = MV – MU, since F = , thus impulse can be written as:
Impulsive force is Force = Impulse/Time. Unit is Newton (N).
The applications of impulsive force
In real life we tend to decrease the effect of the impulsive force by reducing the time taken during collision.
Gravitational force or gravity
Gravity exists due to the earth’s mass and it is acts towards the center of earth. Object falling under the influence of gravity will experience free fall. Assuming no other force acts upon it.
Object experiencing free fall will fall with acceleration; gravity has an approximate value of 10m/s2. The gravitational force acting on any object on earth can be expressed as F=mg. This is also as weight.
For example
Find the gravitational force exerted by the sun on a 79.0kg man located on earth. The distance from the sun to the earth is about 1.50 X 1011 m, and the sun’s mass is
1.99 X 1030kg.
Solution
Fsun = G
= (6.67 X 10-11 Kg-1m3s2)
= 0.413N
Newton’s third law
The action of one body acting upon another body tends to change the motion of the body acted upon. This action is called a force. Because a force has both magnitude and direction, it is a vector quantity, and the previous discussion on vector notation applies.
Newton’s third law is the amount of force which you inflict upon on others will have the same repelling force that act on you as well. Force is exerted on an object when it comes into contact with some other object. Consider the task of driving a nail into a block of wood, for example, as illustrated in the figure 1.7(a). To accelerate the nail and drive it into the block, the hammer must exert a net force on the nail. Newton is a single isolated force (such as the force exerted by the hammer on the nail) couldn’t exist. Instead, forces in nature always exist in pairs. According to Newton, as the nail is driven into the block by the force exerted by the hammer, the hammer is slowed down and stopped by the force exerted by the nail.
Newton described such paired forces with his third law: Whenever one object exerts a force on a second object, the second exerts an equal and opposite force on the first.
This law, which is illustrated in figure 1.7(b), state that a single isolated force can’t exist. The force F12 exerted by object 1 on object 2 is sometimes called the action force, and the force F12 exerted by object 2 on object 1 is called the reaction force. In reality, either, either force can be labeled the action or reaction force. The action force is equal in magnitude to the reaction force and opposite in direction. In all cases, the action and reaction forces act on different objects.
For example, the force acting on a freely falling projectile is the force exerted by earth on the projectile, Fg, and the magnitude of this force is its weight mg. The reaction to force Fg is the force exerted by the projectile on earth, Fg = -Fg. The reaction force Fg must accelerate the earth towards the projectile, just as the action force Fg accelerates the projectile towards the earth. Because the earth has such a large mass and its acceleration due to this reaction forces is negligibly small.
Figure 1.7: Newton’s third law. (a) The force exerted by the hammer on the nail is equal in magnitude and opposite in direction to the force exerted by the nail on the hammer. (b) The force F12 exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force F21 exerted by object 2 on object 1.
Newton’s third law constantly affects our activities in everyday life. Without it, no locomotion of any kind would be possible, whether on foot, on a bicycle, or in a motorized vehicle. When walking, we exert a frictional force against the ground. The reaction force of the ground against our foot propels us forward. In the same way, the tired on a bicycle exert a frictional force against the ground, and the reaction of the ground pushes the bicycle forward. This is called friction plays a large role in such reaction forces.
Figure 1.8:
In the figure 1.8, when a force pushes on an object, the object pushes back in the opposite direction. The force of the pushing back is called the reaction force. This law explains why we can move a rowboat in water. The water pushes back on the oar as much as the oar pushes on the water, which moves the boat. The law also explains why the pull of gravity doesn’t make a chair crash through the floor; the floor pushes back enough to offset gravity. When you hit a baseball, the bat pushes on the ball, but the ball also on the bat.
Figure 1.9
Question Two
Research and illustrate the various characteristics of “Damped Oscillations”, your answer should also include graphical display of these characteristic.
Answer
In the real life, the vibrating motion can be taken place in ideal systems that are oscillating indefinitely under the action of a linear restoring force. In many realistic system, resistive forces, such as friction, are present and retard the motion of the system. Consequently, the mechanical energy of the system diminishes in time, and the motion is described as a damped oscillation.
Thus, in all real mechanical systems, forces of friction retard the motion, so the systems don’t oscillate indefinitely. The friction reduces the mechanical energy of the system as time passes, and the motion is said to be damped.
In the figure 2.0, shock absorbers in automobiles are one practical application of damped motion. A shock absorber consists of a piston moving through a liquid such as oil. The upper part of the shock absorber is firmly attached to the body of the car. When the car travels over a bump in the road, holes in the piston allow it to move up and down in the fluid in a damped fashion.
(b)
Figure 2.0: (a) A shock absorber consists of a piston oscillating in a chamber filled with oil. As the piston oscillates, the oil is squeezed through holes between the piston and the chamber, causing a damping of the piston’s oscillations. (b) One type of automotive suspension system, in which a shock absorber is placed inside a coil spring at each wheel.
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Damped motion varies with the fluid used. For example, if the fluid has a relatively low viscosity, the vibrating motion is preserved but the amplitude of vibration decreases in time and the motion ultimately ceases. This process is known as under damped oscillation. The position vs. time curve for an object undergoing such as oscillation appears in active figure 2.1. In the figure 2.2 compares three types of damped motion, with curve (a) representing underdamped oscillation. If the fluid viscosity is increased, the object return rapidly to equilibrium after it is released and doesn’t oscillate. In this case the system is said to be critically damped, and is shown as curve (b) in the figure 2.2. The piston return to the equilibrium position in the shortest time possible without once overshooting the equilibrium position. If the viscosity is greater still, the system is said to be overdamped. In this case the piston returns to equilibrium without ever passing through the equilibrium point, but the time required to reach equilibrium is greater than in critical damping. As illustrated by curve (c) in figure 2.2.
Active figure 2.1: A graph of displacement versus time for an under damped oscillator. Note the decrease in amplitude with time.
Figure 2.2: Plots of displacement versus time for (a) an under damped oscillator, (b) a critically damped oscillator, and (c) an overdamped oscillator.
Damped oscillation is proportional to the velocity of the object and acts in the direction opposite that of the object’s velocity relative to the medium. This type of force is often observed when an object is oscillating slowly in air, for instance, because the resistive force can be expressed as R = -bv, where b is a constant related to the strength of the resistive force, and the restoring force exerted on the system is -kx, Newton’s second law gives us
= -kx – bv = max
-kx – b = m ~(i)
The solution of this differential equation requires mathematics that may not yet be familiar to you, so it will simply be started without proof. When the parameters of the system are such that b < so that the resistive force is small, the solution to equation is
X = ( Ae-(b/2m)t) cos(wt + ) ~(ii)
Where the angular frequency of the motion is
= ~(iii)
The object suspended from the spring experience both a force from the spring and a resistive force from the surrounding liquid. Active figure 2.1 shows the position as a function of time for such a damped oscillator. We see that when the resistive force is relatively small, the oscillatory character of the motion is preserved but the amplitude of vibration decreases in time and the motion ultimately creases, this system is known as an underdamped oscillator. The dashed blue lines in active figure 2.1, which form the envelope of the oscillatory curve, represent the exponential factor that appears in equation (ii). The exponential factor shows that the amplitude decays exponentially with time.
It is convenient to express the angular frequency of vibration of a damped system (iii) in the form
=
Where = √k/m represents the angular frequency of oscillation in the absence of a resistive force (the undamped oscillator). In other words, when b=o, the resistive force is zero and the system oscillates with angular frequency, called the natural frequency. As the magnitude of the resistive force increases, the oscillations dampen more rapidly. When b reaches a critical value bc,so that bc/2m = , the system does not oscillate and is said to be critically damped. In this case, it returns to equilibrium in an exponential manner with time, as in figure 2.2.
Question Three:
Simple Harmonic Motion (SHM) is a dynamical system typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke’s Law. The motion is sinusoidal in time and demonstrates a single resonant frequency.
What is the relationship between the tension and weight in the system?
What is Hooke’s law when applied to the system?
Answer
Oscillation of motion is has one set of equations can be used to describe and predict the movement of any object whose motion is simple harmonic. The motion of a vibrating object is simple harmonic if its acceleration is proportional to its displacement and its acceleration and displacement are in opposite direction.
The second bullet point mean that are acceleration, and therefore the resultant force, always acts towards the equilibrium position, where the displacement is zero.
Common examples of simple harmonic motion include the oscillations of a simple pendulum and those of a mass suspended vertically on a spring.
The diagram shows the size of the acceleration of a simple pendulum and a mass on a spring when they are given a small displacement, x, from the equilibrium position.
Figure 3.0
In the figure 3.0, the numerical value of the acceleration is equal to a constant multiplied by the displacement, showing that acceleration is proportional to displacement. Then, the negative value of the acceleration shows that it is in the opposite direction to the displacement, since acceleration and displacement are both vector quantities.
Simple harmonic in a spring
If you hang a mass from a spring, the mass will stretch the spring a certain amount and then come to rest. It is established when the pull of the spring upward on the mass is equal to the pull of the force of gravity downward on the mass. The system, spring and mass, is said to be in equilibrium when that condition is met.
If the mass is up or down from the equilibrium position and release it, the spring will undergo simple harmonic motion caused by a force acting to restore the vibrating mass back to the equilibrium position. That force is called the restoring force and it is directly proportional to magnitude of the displacement and is directed opposite the displacement. The necessary condition for simple harmonic motion is that a restoring force exists that meets the conditions stated symbolically as Fr = -kx, where k is the constant of proportionality and x is the displacement from the equilibrium position. The minus sign, as usual, indicates that Fr has a direction opposite that of x.
For example
Figure 3.1
The crank rotates with angular velocity w. Then, the slide will slide between P1 and P.
V2 = W2 (P2-X2)
P = Amplitude or maximum point.
V= Velocity of the slider.
X = Distance from centre point due to velocity, v.
W = Angular velocity of crank.
= 2Ï€f
f =
= 1/T
a = -w2x
Simple pendulum
A simple pendulum is just a heavy particle suspended from one end of an inextensible, weightless string whose other end in fixed in a rigid support, this point being referred to as the point of suspension of the pendulum.
Obviously, it is simply impossible to obtain such an idealized simple pendulum. In actual practice, we take a small and heavy spherical bob tied to a long and fine silk thread, the other end of which passes through a split cork securely clamped in a suitable stand, the length (â„“) of the pendulum being measured from the point of suspension to the centre of mass of the bob.
In the figure 3.2, let S be the point of suspension of the pendulum and 0, the mean or equilibrium position of the bob. On taking the bob a little to one side and then gently releasing it, the pendulum starts oscillating about its mean position, as indicated by the dotted lines.
At any given instant, let the displacement of the pendulum from its mean position SO into the position SA is θ. Then, the weight mg of the bob, acting vertically downwards, exerts a torque or moment – mg/sin θ about the point of suspension, tending to bring it back to its mean position, the negative sign of the torque indicating that it is oppositely directly to the displacement (θ).
Figure 3.2
If d2θ/dt2 be the acceleration of the bob, towards 0, and I its M.I about the point of suspension (S), the moment of the force or the torque acting on the bobn is also equal to I.d2θ/dt2.
I = -mgℓsinθ
If θ is small, the amplitude of oscillation be small, we may neglect all other terms except the first and take sin θ = θ.
I = -mgℓθ,
Whence, =
Since M.I of the bob about the point of suspension (S) is mâ„“2. We have
= = = µθ,
Where = µ
The acceleration of the bob is thus proportional to its angular displacement θ and is directed towards its mean position 0. The pendulum thus executes a simple harmonic motion and its time period is given by
T = 2Ï€ = 2Ï€ = 2Ï€
It being clearly understood that the amplitude of the pendulum is small. The displacement here being angular, instead of linear, it is obviously an example of an angular simple harmonic motion.
Hooke’s law
Vibration motion is an object attached to a spring. We assume the object moves on a frictionless horizontal surface. If the spring is stretched or compressed a small distance x from its equilibrium position and then released, it exerts a force on the object as shown in figure 3.3. From experiment the spring force is found to obey the equation
F = -kx ~(iv)
Where x is the displacement of the object from its equilibrium position (x=0) and k is a positive constant called the spring constant. This force law for springs is known as Hooke’s law. The value of k is a measure of the stiffness of the spring. Stiff springs have large K value, and soft springs have small K value.
In the equation (iv), the negative sign mean that the force exerted by the spring is always directed opposite the displacement of the object. When the object is to the right of the equilibrium position, as in figure 3.3 (a), x is positive and F is negative. This means that force is the negative direction, to the left. When the object is to the left of equilibrium position, as in figure 3.3 (c), x is negative and F is positive, indicating that the direction the force is to the right. Of course, when x = 0, as in figure 3.3 (b), the spring is unstretched and F =0. Because the spring force always acts toward the equilibrium position, it is some time called a restoring force. A restoring force always pushes or pulls the object toward the equilibrium position.
The process is then repeated, and the object continues to oscillate back and forth over the same path. This type of motion is called simple harmonic motion. Simple harmonic motion occurs when the net force along the direction of motion obeys Hooke’s law – When the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point.
Figure 3.3: The force exerted by a spring on an object varies with the displacement of the object from the equilibrium position, x=0. (a) When x is positive (the spring is stretched). (b) When x is zero (the spring is unstretched), the spring force is zero, (c) When x is negative (the spring is compressed), the spring force is to the right.
Conclusion
As my conclusion, Newton’s law was a very useful in nowadays because it is can use the 3 type of law to prevent any accidents in now generation.
First’s law is states that a force must be applied to an object in order to change its velocity. Second’s law is acceration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. Third’s law is whenever we push on something, it pushes back with equal force in the opposite direction.
Second, harmonic oscillation is a type of forced and damped oscillation that is amplitude of a real swinging pendulum or oscillating spring decrease slowly with time until the oscillation stop altogether. This decay of amplitude as a function of time is called damping.
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