如题所述
点A运动轨迹是摆线,
参数方程为:x=r(t-sint),
y=r(1-cost),
dx/dt=r(1-cost),
(dx/dt)^2=r^2(1-cost)^2,
dy/dt=-rsint,
(dy/dt)^2=r^2(sint)^2,
S转一圈为摆线一拱长,
s=∫[0,2π]√[(dy/dt)^2+(dx/dt)^2]dt
=∫[0,2π]r√[(sint)^2+(cost)^2-2cost+1]dt
=∫[0,2π]r√[2(1-cost)dt
=r∫[0,2π]√(sint/2)^2dt
=4r∫[0,π]sint/2d(t/2)
=4r(-cost/2)[0,π]
=8r,
故圆滚动一周时点A运动路程为8r.
参数方程为:x=r(t-sint),
y=r(1-cost),
dx/dt=r(1-cost),
(dx/dt)^2=r^2(1-cost)^2,
dy/dt=-rsint,
(dy/dt)^2=r^2(sint)^2,
S转一圈为摆线一拱长,
s=∫[0,2π]√[(dy/dt)^2+(dx/dt)^2]dt
=∫[0,2π]r√[(sint)^2+(cost)^2-2cost+1]dt
=∫[0,2π]r√[2(1-cost)dt
=r∫[0,2π]√(sint/2)^2dt
=4r∫[0,π]sint/2d(t/2)
=4r(-cost/2)[0,π]
=8r,
故圆滚动一周时点A运动路程为8r.
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