题目内容
如图所示,绝缘传送带与水平地面成37°角,倾角也是37°的绝缘光滑斜面固定于水平地面上且与传送带良好对接,轻质绝缘弹簧下端固定在斜面底端。皮带传动装置两轮轴心相L="6" m,B、C分别是传送带与两轮的切点,轮缘与传送带之间不打滑。现将质量m=0.1kg、电荷量q="+2×" 10-5 C的工件(视为质点,电荷量保持不变)放在弹簧上,用力将弹簧压缩至A点后由静止释放,工件滑到传送带端点B时速度v0= 8m/s,AB间的距离s=1m,AB间无电场,工件与传送带间的动摩擦因数μ=0.25。(g取10m/s2。sin37°=0.6,cos37°=0.8)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/2014082500275271013894.jpg)
(1)求弹簧的最大弹性势能;
(2)若皮带传动装置以速度v顺时针匀速转动,且v可取不同的值(安全运行的最大速度为10 m/s),在工件经过B点时,先加场强大小E=4×104 N/C,方向垂直于传送带向上的均强电场,0.5s后场强大小变为E'="1.2" ×105 N/C,方向变为垂直于传送带向下。工件要以最短时间到达C点,求v的取值范围;
(3)若用Q表示工件由B至C的过程中和传送带之间因摩擦而产生的热量,在满足(2)问的条件下,请推出Q与v的函数关系式。
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/2014082500275271013894.jpg)
(1)求弹簧的最大弹性势能;
(2)若皮带传动装置以速度v顺时针匀速转动,且v可取不同的值(安全运行的最大速度为10 m/s),在工件经过B点时,先加场强大小E=4×104 N/C,方向垂直于传送带向上的均强电场,0.5s后场强大小变为E'="1.2" ×105 N/C,方向变为垂直于传送带向下。工件要以最短时间到达C点,求v的取值范围;
(3)若用Q表示工件由B至C的过程中和传送带之间因摩擦而产生的热量,在满足(2)问的条件下,请推出Q与v的函数关系式。
(1)
(2)
(3) ![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752773758.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752726586.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752757868.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752773758.png)
试题分析:(1) 从A到B的过程中,由机械能守恒定律有
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250027527881103.png)
代入数据可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752726586.png)
(2)工件经过B点运动
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752820514.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752835758.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752851719.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752867668.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752882820.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250027528981090.png)
当场强大小变为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752913346.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752929405.png)
根据牛顿第二定律有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250027529601242.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752976653.png)
工件沿传送带发生的位移
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752991766.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002753023775.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002753038687.png)
所以当传送带以
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752757868.png)
(3) 由于第一个过程中摩擦力为零,所以只在第二个过程中产生热量,
在第二个过程中经历的时间为
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002753085901.png)
传送带在
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002753101327.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002753116521.png)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/201408250027531471240.png)
代入数据可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140825/20140825002752773758.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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