Physics

Question

16. A 6 kg block is falling toward a spring located 1.8 m below. If it has a speed of 4 m/s at first,
what will be the maximum compression of the spring? Assume the spring constant is 2000 N/m
[6 marks][1 mark for Comm]‚Äč

1 Answer

  • Answer:

    Maximum compression of the spring: [tex]\rm 0.423\; m[/tex].

    Explanation:

    Let [tex]x[/tex] be the maximum compression, in meters, of the spring.

    What's the initial kinetic energy of this block?

    • The initial velocity [tex]v[/tex] of the block is [tex]\rm 4\;m\cdot s^{-1}[/tex].
    • The mass of the block is [tex]\rm 6\;kg[/tex].
    • Initial [tex]\displaystyle \mathrm{KE} = \frac{1}{2} \;m\cdot v^{2} = \rm \frac{1}{2} \times 6\times 4^{2} = 48\;J[/tex].

    What's the final kinetic energy of this block?

    • The block would have come to a stop by the time the spring reaches its maximum compression. [tex]v = 0[/tex]. As a result, the final kinetic energy of the block shall also be zero.

    Let the initial height of the block be zero.

    The height of the block will be [tex]\rm -1.8\;m[/tex] by the time it reaches the top of the spring. However, the height of the block will become even more negative as the spring compresses. If the maximum compression of the spring is [tex]x[/tex] meters, the height of the block will be [tex]-(1.8 + x)[/tex] meters by the time the spring reaches its maximum compression.

    The change in the height of the block will thus be [tex]-(1.8 + x)\;\mathrm{m}[/tex] relative to the initial position of the block. The change in the block's potential energy due to gravity will be:

    [tex]\Delta \mathrm{GPE} = m\cdot g\cdot \Delta h = 6\times 9.8 \times (-(1.8 + x)) = -105.948- 58.86\;x[/tex].

    The mechanical energy of an object is the sum of its kinetic energy and its potential energy. What will be the change in the mechanical energy of the block?

    [tex]\text{Change in Mechanical Energy} \\ = \text{Change in KE} + \text{Change in PE} \\ = (0 - 48) + ((-105.948- 58.86\;x) - 0)\\ = (-153.948 - 58.86\;x)\; \text{J}[/tex].

    The block has lost [tex](153.948 + 58.86\;x)\; \text{J}[/tex] of energy as it travels from its initial position to the place where the spring is most compressed. Energy conserves. Those energies are transferred to the spring. Assume that there's no energy loss in this process. All those [tex](153.948+ 58.86\;x)\; \text{J}[/tex] would have become the elastic potential energy of the spring.

    On the other hand, it is assumed that the spring is compressed by a distance of [tex]x[/tex] meters. By Hooke's Law, the elastic potential energy of the spring will be:

    [tex]\displaystyle \text{Elastic Potential Energy} = \frac{1}{2} \; k\cdot x^{2} = \frac{1}{2}\times 2000 \times x^{2} = 1000\; x^{2}\;\text{J}[/tex].

    There are thus two expressions for the Elastic Potential Energy of the spring:

    • [tex](153.948+ 58.86\;x)\; \text{J}[/tex] from the conservation of energy, and
    • [tex]1000\; x^{2}\;\text{J}[/tex] by Hooke's Law.

    The two expression shall be equivalent. Equate the two expressions and solve for the compression of the spring, [tex]x[/tex].

    [tex]153.948+ 58.86\;x = 1000\; x^{2}[/tex].

    [tex]1000\; x^{2}- 58.86\;x - 153.948 = 0[/tex].

    [tex]\displaystyle x_{1} = \frac{-(-58.86) +\sqrt{(-58.86)^{2} - 4\times 1000 \times (-153.948)}}{2\times 1000} = \rm 0.423[/tex].

    [tex]\displaystyle x_{2} = \frac{-(-58.86) +\sqrt{(-58.86)^{2} - 4\times 1000 \times (-153.948)}}{2\times 1000} = -0.364[/tex].

    Consider the expression for the height of the block when it is at the top of the spring and when the spring reaches maximum compression:

    • Height of the block at the top of the spring: [tex]\rm -1.8\;m[/tex];
    • Height of the block when the spring reaches maximum compression: [tex]-(1.8 + x) \; \text{m}[/tex]. This height will be [tex]\rm -2.2\; m[/tex] if [tex]x_{1}[/tex] is correct, and [tex]\rm -1.4\; m[/tex] if [tex]x_{2}[/tex] is correct. The block compresses the spring and will be below its initial position at the top of the spring. Therefore, [tex]x > 0[/tex] and only [tex]x_{1}[/tex] is correct.

    The maximum compression of the spring will be 0.423 meters.

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