Math Problem Statement

Productive Task

A particle is moving along a horizontal line in simple harmonic motion. The displacement of the particle at time t is given by:

x(t) = 5 (sin (2t))5

Find the velocity [w(t) = x'(t)] of the

particle at time t =

π /4

Solution

To find the velocity of the particle at time t=π4t = \frac{\pi}{4}, we first need to differentiate the displacement function x(t)=5(sin(2t))5x(t) = 5 (\sin(2t))^5 to find w(t)=x(t)w(t) = x'(t).

Step 1: Differentiate x(t)x(t)

Using the chain rule, we differentiate x(t)x(t):

x(t)=5(sin(2t))5x(t) = 5 (\sin(2t))^5

Let u=sin(2t)u = \sin(2t). Then, x(t)=5u5x(t) = 5u^5. Using the chain rule, we have:

dxdt=55u4dudt\frac{dx}{dt} = 5 \cdot 5u^4 \cdot \frac{du}{dt}

Now, calculate dudt\frac{du}{dt}:

dudt=cos(2t)2=2cos(2t)\frac{du}{dt} = \cos(2t) \cdot 2 = 2\cos(2t)

Now substituting back:

dxdt=25u4(2cos(2t))=50(sin(2t))4cos(2t)\frac{dx}{dt} = 25u^4 \cdot (2\cos(2t)) = 50 (\sin(2t))^4 \cos(2t)

Thus, the velocity function w(t)w(t) is:

w(t)=50(sin(2t))4cos(2t)w(t) = 50 (\sin(2t))^4 \cos(2t)

Step 2: Evaluate at t=π4t = \frac{\pi}{4}

Now, substitute t=π4t = \frac{\pi}{4}:

w(π4)=50(sin(2π4))4cos(2π4)w\left(\frac{\pi}{4}\right) = 50 \left(\sin\left(2 \cdot \frac{\pi}{4}\right)\right)^4 \cos\left(2 \cdot \frac{\pi}{4}\right)

Calculating the sine and cosine:

sin(π2)=1,cos(π2)=0\sin\left(\frac{\pi}{2}\right) = 1, \quad \cos\left(\frac{\pi}{2}\right) = 0

Thus,

w(π4)=50(1)40=0w\left(\frac{\pi}{4}\right) = 50 \cdot (1)^4 \cdot 0 = 0

Conclusion

The velocity of the particle at t=π4t = \frac{\pi}{4} is:

w(π4)=0w\left(\frac{\pi}{4}\right) = 0

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

  1. How do you derive the displacement function for a different simple harmonic motion?
  2. What are the characteristics of simple harmonic motion?
  3. Can you explain the physical significance of velocity being zero in this context?
  4. How would you find the acceleration of the particle at t=π4t = \frac{\pi}{4}?
  5. What changes if the amplitude or frequency of the motion is altered?

Tip: When dealing with derivatives in physics, always remember to keep track of your units for clarity!

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Simple Harmonic Motion

Formulas

x(t) = 5 (sin(2t))^5
w(t) = x'(t)

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12