Simple Harmonic Motion-should be easy

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In summary, the particle's position can be represented by the expression x = 40 cos(45 t + 8), where t is given in seconds. The angular frequency is 45 rad/sec, the period is 0.139 seconds, and the amplitude is 40 cm. When t = 0.05 seconds, the particle's position is -27 cm. To find its velocity, multiply the original equation by -45 and solve for t = 0.05 seconds. To find its acceleration, multiply the original equation by 2025 and solve for t = 0.05 seconds. The maximum speed attained by the particle occurs when the acceleration is equal to 0, which can be found by setting the acceleration equation
  • #1
mattmannmf
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A particle's position in centimeters is given by the expression x = 40 cos(45 t + 8), where t is given in seconds.

angular frequency= 45 rad/sec
period= .139 s
amplitude= 40 cm

Heres the part i can't get but seems so simple...

where is the particle at t= .05 sec?

what is its velocity?

what is its accleration?

what is the maximum speed attained by the particle?

So what i figured on doing is just plugging .05 into my original equation x = 40 cos(45 t + 8) to solve for distance...where t= .05, distance= 39.36...which came out to be wrong for some reason...(all is in cm)
then for the rest i just add a "-w" infront of the equation for velocity
and a "-w^2" for acceleration and then solve when a=0, velocity has a max or min..

any ideas?
 
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  • #2
First issue, the argument of the cosine is in radians--make sure to set your calculator for radians. I got something like -27.

See if that helps with the remaining questions which are modified by -w and w^2 as you note.
 
  • #3


I would like to clarify a few things about Simple Harmonic Motion. First, the equation provided is not a complete representation of simple harmonic motion. It is only valid for a particle moving in a straight line with simple harmonic motion, and it assumes the particle starts at its equilibrium position (x=0) at t=0. Additionally, the given expression does not have units, which are important in understanding the physical meaning of the values.

To answer the questions, we can use the given equation and the principles of simple harmonic motion. At t=0.05 s, the particle's position can be found by plugging in t=0.05 into the equation:
x = 40 cos(45*0.05 + 8) = 39.36 cm. This is the correct answer, so it seems like there may have been an error in the calculation.

To find the velocity at t=0.05 s, we can use the derivative of the position equation:
v = -40*45*sin(45*0.05 + 8) = -179.93 cm/s. The negative sign indicates that the particle is moving in the opposite direction of the positive x-axis.

Similarly, the acceleration at t=0.05 s can be found by taking the derivative of the velocity equation:
a = -40*45^2*cos(45*0.05 + 8) = -8096.85 cm/s^2. Again, the negative sign indicates that the acceleration is in the opposite direction of the positive x-axis.

The maximum speed attained by the particle can be found by setting the velocity equation equal to zero and solving for t:
0 = -40*45*sin(45t + 8)
sin(45t + 8) = 0
45t + 8 = 0 or 45t + 8 = π
t = -8/45 or t = (π-8)/45
Since t cannot be negative, the maximum speed occurs at t = (π-8)/45 s. Plugging this value into the velocity equation gives us the maximum speed:
v = -40*45*sin(45*(π-8)/45 + 8) = 40 cm/s. This is the same as the amplitude, which makes sense since the particle reaches its maximum speed at the equilibrium position.

In summary, simple harmonic motion is a type of periodic motion where
 

Related to Simple Harmonic Motion-should be easy

What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion where an object moves back and forth along a straight line, such that its acceleration is directly proportional to its displacement from a fixed point. This motion is often seen in objects attached to a spring or pendulum.

What is the equation for simple harmonic motion?

The equation for simple harmonic motion is x = A*cos(ωt + φ), where x is the displacement from the equilibrium position, A is the amplitude of the motion, ω is the angular frequency, and φ is the phase angle.

What is the period of simple harmonic motion?

The period of simple harmonic motion is the time it takes for one complete cycle of the motion. It is given by T = 2π/ω, where ω is the angular frequency.

What is the difference between simple harmonic motion and uniform circular motion?

In simple harmonic motion, the object moves back and forth along a straight line, while in uniform circular motion, the object moves in a circle with constant speed. Additionally, in simple harmonic motion, the acceleration is directly proportional to the displacement, whereas in uniform circular motion, the acceleration is directed towards the center of the circle.

What are some real-life examples of simple harmonic motion?

Some real-life examples of simple harmonic motion include the motion of a mass attached to a spring, the swinging of a pendulum, and the vibrations of a guitar string.

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