physics problem?

[匿名]

2015-12-09 8:58 am

A particle starts from rest at top of a solid hemisphere as shown in Fig Q10.The hemisphere has a radius R and is fixed on the horizontal ground.
(a)If a solid ball of radius r rolls down the surface of the hemisphere without slipping,find the angle θ,where the ball leaves the surface of the hemisphere. The moment of inertia of a solid ball is I=(2/5)mr^2

回答 (1)

麻辣

2015-12-10 11:36 am

✔ 最佳答案

(1) v = ? for c = r + R

E1 = Rotating energy

= 0.5*I*w^2

= 0.5 * 2/5 * mr^2 * [v/(R+r)]^2

= m/5 * [r/(R+r)]^2 * v^2

E2 = Kinetic energy

= 0.5 * m * v^2

E3 = Potential energy

= m*g*(r+R)(1-cosθ)

E3 = E1 + E2

=> v^2 = g(r+R)(1-cosθ)/0.5+0.2(r/r+R)^2

=> v^2 = 10g(1-cosθ)c^3/(5c^2+2r^2) ... (a)

(2) θ = ? for Ball leaving

Fr = Centrifugal force * cosθ

= m*v^2*cosθ/c

= m*g

=> v^2 = g*c/cosθ = Eq.(a)

=> k = (5c^2 + 2r^2)/10c^2

= (1-cosθ)cosθ

= cosθ - (cosθ)^2

=> (cosθ)^2 - cosθ + k = 0

=> cosθ = [1 + √(1-4k)]/2

=> θ = acos{[1 + √(1-4k)]/2}

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