Question

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A spring with spring constant 15 N/m hangs from the ceiling. A ball is attached to the spring and allowed to come to rest. It is then pulled down 6.0 cm and released. If the ball makes 30 oscillations in 20 s, what are its (a) mass and (b) maximum speed?​

Answer 1

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Part (A)

   $m = 0.169 \ kg$

Part (B)

  $|v_{max} |= 0.5653 \ m/s$

Explanation:

From the question we are told that

    The  spring constant is  $k = 14 \ N/m$

     The  maximum extension of the spring is  $A = 6.0 \ cm = 0.06 \ m$

     The number of oscillation is  $n = 30$

      The  time taken is  $t = 20 \ s$

Generally the the angular speed of this oscillations is mathematically represented as

           $w = \frac{2 \pi}{T}$

where T is the period which is mathematically represented as

     $T = \frac{t}{n}$

substituting values

     $T = \frac{20}{30 }$

     $T = 0.667 \ s$

Thus  

       $w = \frac{2 * 3.142 }{ 0.667}$

       $w = 9.421 \ rad/s$

This angular speed can also be represented mathematically as

       $w = \sqrt{\frac{k}{m} }$

=>   $m =\frac{k }{w^2}$

substituting values

      $m =\frac{ 15 }{(9.421)^2}$

      $m = 0.169 \ kg$

In SHM (simple harmonic motion )the equation for velocity is  mathematically represented as

        $v = - Awsin (wt)$

The  velocity is maximum when  $wt = \(90^o) \ or \ 1.5708\ rad$

     $v_{max} = - A* w$

=>   $|v_{max} |= A* w$

=>    $|v_{max} |= 0.06 * 9.421$

=>   $|v_{max} |= 0.5653 \ m/s$

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