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Question

A wave is described by the equation

y=1.0mmsinπx2.0cm-t0.01s

(a) Find the time period and the wavelength?

(b) Write the equation for the velocity of the particles. Find the speed of the particle at x=1.0cm at time t=0.01s

(c) What are the speeds of the particles at x=3.0cm, 5.0cm and 7.0cm at t=0.01s?


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Solution

a Step 1.Given data

Equation of wave y=1.0mmsinπx2.0cm-t0.01s ………… eq1

Step 2. Comparing the given equation to the general equation of the wave

General equation of the wave

y=Asinkx-ωt

where A is amplitude of the wave

k is the wave vector

ω is the angular frequency

Now, comparing the given equation to the general equation of the wave ,we get

A=1mm

k=π2cm-1

ω=100rads-1

Step 3. Finding the time period of the wave

By using the formula of the time period

T=2πω

T=2π100π

T=0.02s

Step 4. Finding the wavelength of the wave

By using the formula of the wavelength

λ=2πk

λ=2ππ×2

λ=4cm

Hence , the time period of the wave is0.02s and wavelength of the wave is4cm

b Step 1. Given data

x=1.0cm

t=0.01s

Step 2. Finding the velocity of wave

If we derivate equation 1 with respect to time then, we get

velocity ,v=dydt

v=-ωAcoskx-ωt ………..eq2

Step 3. Finding the velocity at x=1.0cm and t=0.01s

Now putting the given values in eq2, we get

v=-100πcosπx2-100πt

v=-100πcosπ2-π

v=0

Hence the velocity at x=1.0cmis 0

c Step 1. Given data

x=3cm,5cm,7cm

Step 2. Finding the velocity at x=3cm

Now putting x=3cm in eq2 ,we get

v=-100πcos3π2-π

v=0

Step 3. Finding the velocity at x=5cm

Putting x=5cm in eq2, we get

v=-100πcos5π2-π

v=0

Step 4. Finding the velocity at x=7cm

Putting x=7cm in eq2, we get

v=-100πcos7π2-π

v=0

Hence , velocity at x=3cm,5cm,7cm is 0.


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