In an experiment, students are to calculate the spring constant kof a vertical s

Question

In an experiment, students are to calculate the spring constant kof a vertical spring in a small jumping toy that initially rests ona table. When the spring in the toy is compressed a distance x fromits uncompressed length Lo and the toy is released, the top of thetoy rises to a maximum height h above the point of maximumcompression. The students repeat the experiment several times,measuring h with objects of various masses taped to the top of thetoy so that the combined mass of the toy and added objects is m.The bottom of the toy and the spring each have negligible masscompared to the top of the toy and the objects taped to it.

(a) Derive an expression for the height h in terms of m, x, k, andfundamental constants.

With the spring compressed a distance x=0.020 m in each trial, thestudents obtained the following data for different values of m.

           m (kg)       h (m)
           0.020        0.49
           0.030        0.34
           0.040        0.28
           0.050        0.19
           0.060        0.18

(b)
      i. What quantities should be graphedso that the slope of a best-fit straight line through thedata points can be used to calculate the spring constantk?

      ii. Fill in one or both of the blankcolumns in the table with the calculated volumes of yourquantities, including units

(c) Calculate the numerical value of the spring constant.

(d) Describe a procedure for measuring the height h in theexperiment, given that the toy is only momentarily at the maximumheight.

Explanation / Answer

k=F/x F=ma=mg so (a)k=mg/x (b)use mean values for mg and x to find .04g/.296=1.32N/m=k (c)graph values for mg on y axis distance x on x axis (d)calculate the distance traveled by timing how long theobject is in the air using a stop-motion timer and use that time tofind h.

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