如图*所示,间距为L、足够长的固定光滑平行金属导轨MN、PQ与水平面成θ角,左端M、P之间连接有电流传感器和阻...
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如图*所示,间距为L、足够长的固定光滑平行金属导轨MN、PQ与水平面成θ角,左端M、P之间连接有电流传感器和阻值为R的定值电阻。导轨上垂直停放一质量为m、电阻为r的金属杆ab,且与导轨接触良好,整个装置处于磁感应强度方向垂直导轨平面向下、大小为B的匀强磁场中。在t= 0时刻,用一沿MN方向的力斜向上拉金属杆ab,使之从磁场的左边界由静止开始斜向上做直线运动,电流传感器将通过R的电流i即时采集并输入电脑,可获得电流i随时间t变化的关系图线,电流传感器和导轨的电阻及空气阻力均忽略不计,重力加速度大小为g。
(1)若电流i随时间t变化的关系如图乙所示,求t时刻杆ab的速度υ大小;
(2)在(1)问的情况下,请判断杆ab的运动*质,并求t时刻斜向上拉力的功率P;
(3)若电流i随时间t变化规律为i=Imsint,则在0~T时间内斜向上拉力对杆ab做的功W。
【回答】
【标准解答】(1)由乙图可知,t=t1时刻电路中的感应电流为I1,则t时刻,电流为
i=t ··········································································································· ①(1分)
杆ab切割磁感线产生的感应电动势为
e = BLυ··············································································································· ②(1分)
根据闭合电路欧姆定律有
e = i( R + r ) ······································································································ ③(1分)
由以上三式解得
υ = t ································································································ ④(1分)
(2)由于是常量,所以杆ab是做匀加速直线运动,其加速度大小为
a = = ························································································ ⑤(1分)
设t时刻水平拉力大小为F,根据牛顿第二定律有
F–BiL–mgsinθ=ma ······················································································· ⑥(1分)
又 P=Fυ········································································································· ⑦(1分)
得 P=t2 + [gsinθ + ] t ···························· ⑧(2分)
(3)设位移x=–Acos t,由导数的物理意义可知,有
υ=A · sint ·························································································· ⑨(1分)
由BLυ=i( R + r )及i=Imsint 得
υ=sint ···················································································· ⑩(1分)
可见,杆ab做简谐运动。
所以振幅A= ········································································ (11)(1分)
在0 ~ T时间内,重力做功为W1 =–mg · 2Asinθ········································· (12)(1分)
t=T时刻,杆ab的速度大小
υ= 0 ·········································································································· (13)(1分)
在0~T时间内,整个回路产生的焦耳热为
Q= ()2( R + r ) · T ·············································································· (14)(1分)
安培力对杆ab做的功为
W2 =–Q ······································································································ (15)(1分)
根据动能定理有
W + W1 + W2= – 0 ········································································· (16)(1分)
联立以上四式解得
W= + Im2( R + r )T ················································· (17)(1分)
【思维点拔】本题的关键在于对电流传感器得到的电流i随时间t变化的关系图线的理解,获取信息,从加速度定义来确定运动*质,利用数学知识位移的导函数是速度函数来确定简谐运动的振幅,从而来求重力做功,整个回路产生的焦耳热要用电流的有效值来算,然后借助动能定理加以求解。
知识点:专题四 功和能
题型:综合题
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