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19 -- Part 1: Vibrational Motion

- A Russian cavalry squadron of 50 soldiers were crossing a bridge over the Fontanka River in 1905 when it collapsed.

- Hooke's law can be applied to analyze forces simultaneously walking or riding horses across the stone bridge, which has stood for 81 years.

- The use of radians to describe angles is different from ordinary crowds.

- Russian soldiers crossed the bridge in step.

- The process will be converted into an out in this chapter.

- Linear motion can be studied at either constant velocity or constant acceleration.

- The motion is moving in a circle.
- In the case of circular motion, the di rection of the velocity and acceleration of the objects varies.
- There is a new type of motion in this chapter.
- When you walk, when you talk, and when you listen to music, you experience this motion.

- Your legs and arms swing back and forth when you walk.
- While you walk.

- The goal of this chapter is to investigate this type of motion.

- A spring exerts force on an object that is stretched or compressed.

- The sum of the forces that other objects exert on the cart is zero when it is at rest.
- When the cart is released from this position, it moves back and forth in two different directions.
- A force is put on the vibrating object that tends to return it to the rest position.

- When not disturbed, the place where it resides.

- We used force diagrams and sketches to analyze the motion of the cart-spring system.

- The right side repeats over and over.
- We use motion diagrams, force diagrams, and energy bar charts to analyze the process.

- The energy in between is a combination of the two.

- General motions of motion are described in the observational experiment tables.
- An object moves in one direction and in the opposite direction during a motion called vibrational motion.

- The object is moving at high speed, first in one direction, then the other.
- A component of a restoring force on it by some other object points back to equilibrium when it overshoots.

- It's not always desirable to have vibrating objects.
- Tal buildings must be constructed so that they don't sway too much during earthquakes or on very windy days, and cars must have shock absorbers to absorb the up-and-down shocks that occur on a bumpy road.
- There are tall trees swaying on the surface of a speaker.

- Brian puts a book on a desk and pushes it to the left.
- The book goes through motion.

- Explain what makes Brian say this.
- Explain why this motion doesn't fit our definition of motion.

- We can use amplitude to describe motion.

- The time interval is an important quantity.

- This quantity is from our study of circular motion.

- The time interval for Earth to make one complete rotation on its axis is one day.

- The period is 0.25 s and the Frequency is 4 vib/s.
- The words vibrate and cycle are not units.
- The words cycle and vibration can be removed from the equations.
- The appropriate unit is 1/s or s-1.

What is it that it has?

- You are sitting on a chair.

- The periods and frequencies of rocking should be determined.

- 8.0 s is the Frequency.

- Let's look at the data from an experiment in which a motion detector collects position-versus-time, ve locity-versus-time, and acceleration-versus-time data for a cart vibrating on a spring.

- The slope of the graph changes with the object's position.
- There is a rela tionship between the position and acceleration graphs.
- When the position has its maximum positive value, keematics graphs for its maximum negative value.

- This makes sense.

- The cart slopes due to maximum leftward negative acceleration.

- The velocity-versus-matics graphs are also consistent with the acceleration-versus-time graph.

- Imagine a rotation of the circle around the origin of the system.

- 0 to 0 to 0 and then back to zero.

- The match remains strong if we compare more times.
- We need to replace him with an appropriate function of time.
- The ratio of the angle and clock reading is listed in the table.

- The motion was described by Eq.
- There is a mathematical model of motion.
- A cart attached to a spring undergoes motion that is similar to SHM based on position-versus-time graphs.

- Table 19.3 shows the position of a vibrating object.

- The two objects are shown in the figure when the object is zero.
- The function describes the motions.

- They have the same period but have different values at different times of the day.
- The relationships between the three graphs were cussed out earlier in the section.

- The derivative of the func tion can be used to find the rate of change when you have a function of time.
- We don't show the operations in this book, but we will give you the final results.
- If the position function is given.

- The acceleration of a vibrating object is related to its displacement from equilibrium.

- The acceleration is 7.2 m>s2.

- placement at time zero is proportional to the maximum speed.

- We have so far described simple motion using sketches, motion diagrams, and graphs.

- The light spring exerts force on the cart when stretched.
- The other part of the spring is fixed.

- The displacement and acceleration of an object are related.
- The spring pushes the cart in the positive direction if it is on the negative side of the equilibrium.
- The magnitude of the force that the spring exerts on the cart is greater if the cart is from equilibrium.

- The system's motion can be described as mathematical y as a result of this relation.

- Stop for a second and think about how unusual it is.
- The acceleration of a moving object was the same at any location and did not depend on the position, when we studied linear motion at constant acceleration.
- When we studied constant speed circular motion, the direction of acceleration changed, but the magnitude remained the same.
- The displacement of the object is synchronized with the change in time in magnitude and direction of the object.
- The sum of the forces on an object moving in a circle at constant speed changes direction only when it points toward the center of the circle.
- The magnitude of the forces on an object changing direction during a motion is related to the sum of the forces exerted on it.

- We now know that.
- The longer it takes the cart to go through a full cycle, the more you think the period de pends on the amplitude.
- It's easy to check this by starting the vibra tion with different frequencies.
- The period is essential even if the amplitude is doubled or tripled.
- It is reasonable to think that this is surprising.
- The greater the elastic force on the cart by the spring, the faster it moves to ward off the equilibrium position.
- The two effects compensate for the fact that it moves faster over a longer distance.
- The period is not dependent on the amplitude.

- The period of the vibration could be affected by two other factors.
- The stiffening of the spring will increase the acceleration of the cart.
- The period should be affected by the mass of the cart.
- The second law can be used.
- The first thing we do is start with Eq.

- There is no dependency on the amplitude in this expression.

- Hooke's law states that the spring has zero mass, and that the cart is a point-like object.
- We neglected it.
- Most springs obey Hooke's law if you don't stretch them too far, but it's hard to find a massless spring.
- The above equation agrees with experiments in which the mass of the spring is less than the mass of the attached object.

- We can use it.

- Then, add up the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of the coefficients of The cord should be 0.40 Hz.

- It may not fit in a doorway if the cord is increased to 24 kilograms without changing the spring constant.

- The force approach has been used to analyze the motion of the systems.
- The energy approach can help us understand the process better.
- The cart vibrates at the end of the spring.

- On the way back to its starting position, the cycle repeats.
- The system's total energy is constant.

- This is what follows from the kinematics equations.

- Having the mass in the denominator makes sense when the cart is large.
- The cart will move faster if the spring is stiff.

- The system with the surface of the track and the air was neglected in the above discussion.
- Negative work on the system would be done and gradual decrease of energy would bring the vibrating system to rest.

- Table 19.4 shows the situation.
- The cart is the system.

- The elastic potential energy fluctuates in a repetitive pattern.

- There is a giraffe's legs den.
- One vibrating system in which the motion is very apparent is a pen that behaves like a pendulum.
- Pendulums are used in Grandfather clocks and metronomes.

- The movement of our arms is similar to a pendulum.

- The equilibrium position is for the bob.
- If you pull the bob to the right side and release it, it swings to the left, overshooting the equilibrium position, and then stops at about the same distance to the left as it started on the right.

- The bob stops about where it started when it swings back toward the equilibrium position.

- The string S exerts a force in the direction of the path of the bob.

- When the bob is on the right side of equilibrium, the force exerted by Earth points towards the left side.
- The pendulum bob is always trying to get back to its equilibrium position.

- The idea of equilibrium position and restoring force also apply to the motion of a pendulum.

- The right side was released.
- It stops on the other side of the equilibrium position, which is the same distance to the left as it started on the right.
- After overshooting, it returns to its starting position.

- The energy in between is a combination of the two.

- The motion of the pendulum is similar to the motion of the cart on a spring in that it passes the equilibrium position from two different directions.

- The table shows the effect of bob mass, string length, and amplitude on the pendulum period.

- The angle that the string makes with the vertical direc tion is measured.
- The quantities that have been changed are boldfaced.

- The period depends on the string length.
- It doesn't depend on the mass of the bob.
- If it agrees with the above, let's make a formal derivation of the period.

- Suppose that the pendulum string has a positive and a negative displacement from the vertical to the right and left.
- The force exerted by Earth can be resolved into components.
- The two components give the bob the radial acceleration it needs to stay on the path.
- The restoring force points in a different direction to the direction of the displacement.

- For smal angles, the radian units' value is roughly equal to the sin unit.
- For this moderately large angle, sin and U differ by 2%.

- The period of a pendulum is given by Eq.

- The period of a simple pendulum depends on its length and magnitude.
- It doesn't depend on the mass of the object hanging at its end.
- We discovered the patterns in Table 19.6.
- The pendulum's motion does not depend on the bob's mass.

- The number of steps that is "natural" for a leg to natural to take one step with your take per second while walking: in other words, the natu- right leg (or with your left leg) ap ral swinging Frequency of the leg.
- The leg is treated as close to the real approximate as possible.

- Imagine a leg that is about 1 m long swinging from hip to hip.
- The only thing we need to know is the length of the pendulum, so we can estimate back and forth.

- The period for one swing is 60 s by this number.

- The giraffe's mass is not at the foot, but at the center of the legs, so it can take relatively few steps per second.
- Our result is an estimate.
- hua has short legs and a high walking speed.

- The leg is an example of a physical pendulum, an extended body that swings back and forth about a pivot point.

- The energy approach can be used to analyze the motion of a pendulum.
- In Conceptual Exercise 19.7, we do that.

- Start with the sketches of the tion for the first swing, then at the moment the person process at the required instants, see the left sides of the release the swing while it is at its elevated position; accompanying figures.

- The ropes' pivot points are ignored.
- The ropes don't work on the system because they are always parallel to the child's motion.
- The person who raises the swing to its starting position is the only external object that works on the system.

- The process can be represented with bar charts.
- The first chart shows work done on the system.

- If you don't ignore air resis tance and friction, the child will only return to half their original height after a few swings.

- The air and pivots are included.

- Half of the original potential energy has been converted to internal energy due to air resistance and the ropes' pivots.

- The Measurement device is on your grandfather's pendulum clock.

- In space, astronauts lose body mass because their muscles don't get enough use.
- NASA developed the Body Mass Measurement Device to monitor the mass of astronauts since traditional scales don't function in space.
- The chair is held in place by a spring.
- The chair vibrates when it is pulled back and released, depending on the mass of the chair.

- The BMMD chair has a period of 1.2 s when empty.

- Determine the effective spring constant of the chair's spring, the mass of the astronauts, and the maximum speed of the astronauts if the wave is 0.10 m.

- We chose the chair and the spring as the system for (a) and the lem statement.

- The chair is of interest.

- Assume that the chair's spring obeys approximations.

- Represent the process with force.
- If needed, we determine the effective grams and bar charts.

- The process is represented by a bar chart starting from the state when the chair is farthest from equi librium and ending as the chair and person pass equilibrium.

- 2p is the amount of time it takes to determine the astronauts mass.

- If necessary, use the expressions for the determine the maximum speed as the cart passes through the equilib period of an object attached to a spring or a pendulum.

- Iting cases, etc.

- The value is 0.30 m>s.

- The units are correct.
- A person with a mass of 66 kilogram can have a speed of 0.3 m/s and the magnitudes are reasonable.

- If you and the Imagine that you ski down a slope wearing a ski vest cart are the system, then during the moment of the col ision, and then continue sking on a horizontal surface at the bottom, the system is constant.
- After the col ision, you run into a padded cart.
- We can use impulse-momentum on skis.
- To determine your speed immediately after the col is attached to the other end of the cart, you need a 1280-N/m spring ideas.
- You and the cart have the same energy.
- Just before you hit the cart, your speed is 16 m/s.

- The magnitude of the maximum acceleration can be determined using the second law.

- The first figure is the labeled sketch.
- The speed seems reasonable and the sign matches the original direction of motion.

- The maximum dis chart shows the col placement of the cart from its equilibrium position to the right.

- The units are fine during the instant of col- moving.

- The component of momentum is constant.

- The second figure shows the amount of this process.

- The cart was moving at 12 m/s after you collided with it.
- This is the maximum speed after the collision.

- The effect of friction on vibrating objects was mostly ignored in the preceding sections.
- Without it, a car would continue vibrating on its suspension system for miles after crossing abump on a road and tal buildings would continue to sway even after the wind had died down.

- It is possible to observe the effects of friction on a system.
- The blocks should be vibrating by moving the top of each spring up and down for a short time.
- The block in the air has a constant period and reduces in size.

- A shock absorber is used.

- The block in glycerin isn't complete.

- The design of vehicles and bridges can be affected by damp.
- Mountain bikes have shock absorbers to decrease the force on the tire when it hits a hole in the trail.
- The front fork of the bicycle has shock absorbers in it.
- The oil is pushed through the small hole in the spring to help the spider locate the prey.

- The spring's elastic potential energy is converted into internal energy in the oil.
- The spring returns quickly to its noncompressed position instead of bouncing up and down.

- In some cases, damping is not desirable.
- Spiders spin webs to locate and restrain prey.
- The spider will be able to locate the insect if it strikes the web.

- Consider position- versus-time graphs.

- A system that is weakly damped and one that is over damped are different.

- The equilibrium position was damped.

- After opening the door, the time damper returns it to its closed position.

- The position is the shortest possible time.

- You exert a constant force that did positive work on the child and swung until the swing cable swing.
- There was no noise during this time.

- Y on C went down.

- You push her gently with a different period.
- Air resistance can cause the swinging amplitude to go down.

- We use these patterns to predict the results of the pendulum oscillators.

- The string has five pendulums attached to it.

- The 80 cm pendulums vibrate string back and forth at the same Frequency, which tugs on the other at the same Frequency.

- We think that only the other 80 cm pendulum will eventually swing with large amplitude.
- When an object exerts a force on something that varies in time and does net positive work over time, resonance occurs.
- This work will lead to an increase in the total energy of the system.
- The increase in energy occurs when the natural frequencies of the systems are close to the natural frequencies of the systems with an external driving force.
- The child on the swing showed another example of resonance, when the force on her was close to the swing's natural frequencies.

- There is an even more unusual thing that happens when we observe the pendulums.
- As time passes, the original 80 cm pendulum's amplitude goes to zero and the second 80 cm pendulum's goes to a maximum.
- As the second pendulum's amplitude decreases, the first pendulum begins vibrating again.
- The energy is being transferred from one pendulum to another.

- The concept of energy transfer through resonance can be used to explain many everyday phenomena.

- The natural frequencies of the objects must be close to the bridge.

- In our chapter opening story about the Russian soldiers, we can see an example of energy transfer through resonance.
- The collapse of the bridge was caused by large-amplitude vibrations in the bridge when the marching initiated their steps.
- The positive work done by the soldiers on the bridge made the bridge louder.
- Parts of the bridge were pulled apart as the bridge's amplitude increased.

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