Say you are performing your ball-and-ramp experiment (Lab 10) with two balls tha

Question

Say you are performing your ball-and-ramp experiment (Lab 10) with two balls that look identical in every respect, with the same radius r and mass m. One ball has its mass evenly distributed, with moment of inertia I1 = 2 5mr2. The other ball is hollow inside, with I2 = mr2. Making the usual simplifying assumptions rolling without slipping, no air resistance find equations for R1 and R2, the distance from the table for each ball. The only variables appearing in your equations for R1 and R2 should be h and H.

Explanation / Answer

here,

for ball 1

I1 = 0.4 * m * r^2

w = v/r

let the velocity of ball at table be v

using conservation of energy

m * g * h = 0.5 * m * v^2 + 0.5 * I * w^2

2 * g * h = v^2 + 0.4 * m * r^2 * ( v/r)^2

v = sqrt(14 * h) m/s

time taken to reach floor be t

u0 * t + 0.5 * g * t^2 = H

t = sqrt( H/4.9)

R1 = v * t

R1 = sqrt(14 * h) * sqrt( H/4.9)

R1 = 1.69 * sqrt(h*H)

for ball 2 ,

I1 = m * r^2

w = v/r

let the velocity of ball at table be v

using conservation of energy

m * g * h = 0.5 * m * v^2 + 0.5 * I * w^2

2 * g * h = v^2 + m * r^2 * ( v/r)^2

v = sqrt(9.8 * h) m/s

time taken to reach floor be t

u0 * t + 0.5 * g * t^2 = H

t = sqrt( H/4.9)

R2 = v * t

R2 = sqrt(9.8 * h) * sqrt( H/4.9)

R2 = 1.41 * sqrt(h*H)

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