Waves Physics one

Question

A wire with a mass of 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with a frequency of 60 Hz and with an amplitude at the antinodes of 0.300 cm.

What is the speed of propagation (in m/s )of transverse waves in the wire?

I got this part right the answer is 96

Compute the tension force (in N) in the wire.

I also got this part right, 460.8N



What I need help with is the last two parts, I am not sure how to solve them and I would appreciate any help with setting up the problems/ what equations/formulas to use.

What is the maximum transverse velocity (in m/s) of the particle in the wire? 

What is the maximum transverse acceleration (in m/s²) of the particle in the wire? 

Thank you in advance!

Answer 1

In general, a point undergoing harmonic motion has displacement:

s(t) = A×sin(2πft), where

A is the amplitude,

f is frequency,

t is time.

Its velocity is the derivative of displacement:

v(t) = 2πfA×cos(2πft)

Its acceleration is the derivative of velocity:

a(t) = -4π²f²A×sin(2πft)

Since sin and cos range between -1 and 1, the maximum velocity and acceleration are:

v_max = 2πfA = 2π×60s^-1×0.3m = 113.1 m/s

a_max = 4π²f²A = 4π²×3600s^-2×0.3m = 42636.7 m/s²