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1.
(2021高三上·泰安期中)
某兴趣小组测量一缓冲装置中弹簧的劲度系数,缓冲装置如下图所示,固定在斜面上的透明有机玻璃管与水平面夹角为30°,弹簧固定在有机玻璃管底端。实验过程如下:先沿管轴线方向固定一毫米刻度尺,再将单个质量为200g的钢球(直径略小于玻璃管内径)逐个从管口滑进,每滑进一个钢球,待弹簧静止,记录管内钢球的个数n和弹簧上端对应的刻度尺示数
,数据如下表所示。实验过程中弹簧始终处于弹性限度内。采用逐差法计算每个小球对应的弹簧压缩量,进而计算弹簧劲度系数。
n | 1 | 2 | 3 | 4 | 5 | 6 |
| 21.09 | 18.11 | 15.07 | 12.05 | 9.03 | 6.01 |
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(1)
利用
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3E%CE%94%3C%2Fmi%3E%3Cmsub%3E%3Cmi%3EL%3C%2Fmi%3E%3Cmi%3Ei%3C%2Fmi%3E%3C%2Fmsub%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmsub%3E%3Cmi%3EL%3C%2Fmi%3E%3Cmi%3Ei%3C%2Fmi%3E%3C%2Fmsub%3E%3Cmo%3E%E2%88%92%3C%2Fmo%3E%3Cmsub%3E%3Cmi%3EL%3C%2Fmi%3E%3Cmrow%3E%3Cmi%3Ei%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsub%3E%3Cmo+stretchy%3D%22false%22%3E%28%3C%2Fmo%3E%3Cmi%3Ei%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E++%EF%BC%8C+%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E++%EF%BC%8C+%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo+stretchy%3D%22false%22%3E%29%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
计算弹簧的压缩量:
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3E%CE%94%3C%2Fmi%3E%3Cmsub%3E%3Cmi%3EL%3C%2Fmi%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E9.04%3C%2Fmn%3E%3Cmtext%3Ecm%3C%2Fmtext%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3E%CE%94%3C%2Fmi%3E%3Cmsub%3E%3Cmi%3EL%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E9.08%3C%2Fmn%3E%3Cmtext%3Ecm%3C%2Fmtext%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,
cm,滑进每个钢球对应压缩量的平均值为
cm;(结果保留2位小数)
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(2)
若不计摩擦,重力加速度g取
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmn%3E9.80%3C%2Fmn%3E%3Cmsup%3E%3Cmrow%3E%3Cmtext%3Em%2Fs%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,该弹簧的劲度系数为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmtext%3EN%3C%2Fmtext%3E%3Cmo%3E%2F%3C%2Fmo%3E%3Cmtext%3Em%3C%2Fmtext%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
。(结果保留2位小数)
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