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nignag [31]
2 years ago
7

A mass of 0.5 kg hangs motionless from a vertical spring whose length is 1.10 m and whose unstretched length is 0.50 m. Next the

mass is pulled down to where the spring has a length of 1.30 m and given an initial speed upwards of 1.3 m/s. What is the maximum length of the spring during the motion that follows?
Physics
2 answers:
ser-zykov [4K]2 years ago
5 0

Answer:

The maximum length during the motion is L_{max} = 1.45m

Explanation:

From the question we are told that

           The mass  is  m =0.5 kg

            The vertical spring  length is  L = 1.10m

            The unstretched  length is  L_{un} = 1.30m

          The initial speed is v_i = 1.3m/s

          The new length of the spring L_{new} =  1.30 m

The spring constant k is mathematically represented as

                           k = -\frac{F}{y}

Where F is the force applied  = m * g = 0.5 * 9.8=4.9N

           y is the difference in weight which is   =1.10-0.50=0.6m

The negative sign is because the displacement of the spring (i.e its extension occurs against the force F)

    Now  substituting values accordingly

                    k =  \frac{4.9}{0.6}

                       = 8.17 N/m

The  elastic potential energy is given as E_{PE} = \frac{1}{2} k D^2

  where D is this the is the displacement  

Since Energy is conserved the total elastic potential energy would be

             E_T = initial  \ elastic\ potential \ energy + kinetic \ energy

            E_T = \frac{1}{2} k D_{max}^2 =   \frac{1}{2} k D^2 + \frac{1}{2} mv^2

Substituting value accordingly

                \frac{1}{2} *8.17 *D_{max}^2 =\frac{1}{2} * 8.17*(1.30 - 0.50)^2 + \frac{1}{2} * 0.5 *1.30^2

                4.085 * D_{max}^2 = 3.69

                 D^2_{max} = 0.9033

                D_{max} = 0.950m

So to obtain total length we would add the unstretched length

 So we have

                  L_{max} = 0.950 + 0.5 = 1.45m

                               

               

               

                 

                     

Over [174]2 years ago
5 0

Answer:

Explanation:

The weight of the spring is equal to the restoring force of the spring.

F = mg = k× ΔL

ΔL = change in length of the spring when the mass was first hung on the spring.

Where

Given Lo = 0.5m

L = 1.10m

ΔL = 1.10-0.5 = 0.6m

m = 0.5kg

g = 9.8m/s²

k = mg/ΔL = 0.5×9.8/0.6 = 8.2 N/m

Required to find Lmax

Lmax = Lo + A

A = Amplitude of the oscillation

A = √(ΔLo² + Vo²/ωo²)

ΔLo = 1.30 – 0.5 = 0.8m

Vo = 1.30m/s = initial upward velocity

ωo = initial angular frequency.

ωo = √(k/m) = √(8.2/0.5) = 4.05rad/s

A = √(ΔLo² + Vo²/ωo²)

A = √(0.8² + 1.30²/4.05²) = 0.86m

L = 0.5 + 0.86 = 1.36m

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