Finding acceleration, velocity, and time for simple harmonic motion

In summary, a cheerleader is performing simple harmonic motion with an amplitude of .180m and a frequency of .850Hz. The maximum magnitude of acceleration and velocity can be found using formulas ii and iii. The time required to move from the equilibrium position to a point .120m away can be calculated using formula i. Formula i can also be used to find the speed and acceleration when the pom-pom's coordinate is x=+.090m. However, the energy approach (E = K + U = 1/2 mv^2 + mgh = 1/2 kA^2) can only be used to find velocity, and not displacement or acceleration.
  • #1
andreaumali
2
0

Homework Statement



A cheerleader waves her pom-pom in simple harmonic motion with an amplitude of .180m and a frequency of .850Hz.

a) Find the maximum magnitude of the acceleration and of the velocity.

b) Find the acceleration and speed when the pom-pom's coordinate is x=+.090m.

c) Find the time required to move from the equilibrium position directly to a point .120m away.

d) Which of the quantities asked for in parts (a), (b), and (c) can be found using the energy approach (E = K + U = 1/2 mv^2 + mgh = 1/2 kA^2) and which cannot? Explain.


Homework Equations



i. x=Asin(ωt)

ii. vx,max=ωA

iii. ax,max=-ω2A

where ω=angular frequency, A=amplitude, x=displacement from equilibrium, t=time

K=1/2 mv^2

U=mgh


The Attempt at a Solution



I was able to calculate (a) using formulas ii and iii. vmax=.961m/s; amax=5.13m/s^2

I was also able to calculate (c) using formula i. x=7.16s

However, I am unsure of how to find (b).

I think (d) is velocity but I'm not sure if from velocity you'd be able to get x and acceleration?
 
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  • #2
For part b you know ω, A, and x. You should be able to solve for and find t.

I guess one way to solve "b" is by using your equation(i), you can find speed and acceleration by taking the 1st and 2nd derivative wrt t.
 

1) How do you find the acceleration for simple harmonic motion?

The acceleration for simple harmonic motion can be found by using the formula a = -ω²x, where ω is the angular frequency and x is the displacement from the equilibrium position. This formula is derived from the equation for acceleration in SHM, a = -kx/m, where k is the spring constant and m is the mass of the object.

2) What is the relationship between velocity and displacement in simple harmonic motion?

The relationship between velocity and displacement in simple harmonic motion is that they are always out of phase by a quarter of a cycle. This means that when the displacement is at its maximum, the velocity is zero, and vice versa. This can be represented by the equation v = -ωx, where v is the velocity and ω is the angular frequency.

3) How can you calculate the period of simple harmonic motion?

The period of simple harmonic motion can be calculated by using the formula T = 2π/ω, where T is the period and ω is the angular frequency. The period represents the time it takes for one full cycle of the motion to occur.

4) Can you find the velocity and acceleration at any point in time during simple harmonic motion?

Yes, both velocity and acceleration can be found at any point in time during simple harmonic motion. This is because they are both functions of time, and their values can be calculated using the equations v = -ωx and a = -ω²x, where x is the displacement from the equilibrium position.

5) How does the amplitude affect the motion of an object in simple harmonic motion?

The amplitude, or the maximum displacement from the equilibrium position, does not affect the motion of an object in simple harmonic motion. This is because the equations for velocity and acceleration, v = -ωx and a = -ω²x, are independent of the amplitude. Therefore, changing the amplitude will not change the velocity or acceleration of the object at any point in time.

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