关于高数曲率的一道应用题汽车连同载重共5t,在抛物线桥上行驶,速度为26.1km/h,桥的跨度为10m,桥的矢高为0.25m,求汽车越过桥顶时对桥的压力.
2019-04-14
关于高数曲率的一道应用题
汽车连同载重共5t,在抛物线桥上行驶,速度为26.1km/h,桥的跨度为10m,桥的矢高为0.25m,求汽车越过桥顶时对桥的压力.
优质解答
F = mg - mv²/r
在抛物线顶点处r等于该点曲率圆的半径,也就是曲率的倒数
曲率K = |y''|/√(1+y'²)³
以顶点为原点,顶点处切线为x轴,则该抛物线方程是y = -ax²,且满足x=5时,y = -0.25,则a = 0.01,算出y' = -0.02x,y'' = -0.02
则在x=0处K = 0.02/1 = 0.02
r = 1/K = 50
代入F得F = 5000 * 9.8 - 5000 * (26.1/3.6)²/50 = 43743.8(N)
F = mg - mv²/r
在抛物线顶点处r等于该点曲率圆的半径,也就是曲率的倒数
曲率K = |y''|/√(1+y'²)³
以顶点为原点,顶点处切线为x轴,则该抛物线方程是y = -ax²,且满足x=5时,y = -0.25,则a = 0.01,算出y' = -0.02x,y'' = -0.02
则在x=0处K = 0.02/1 = 0.02
r = 1/K = 50
代入F得F = 5000 * 9.8 - 5000 * (26.1/3.6)²/50 = 43743.8(N)