Question

The figure shows an overhead view of a ring that can rotate about its center like a merry-go-round. Its outer radius R2 is 0.9 m, its inner radius R1 is R2/2, its mass M is 8.0 kg, and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of 8.5 rad/s with a cat of mass m = M/4 on its outer edge, at radius R2. By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius R1?

Answer 1

Here no figure is given ,bt according the question, figure is imagined and start,

Angular kinetic energy of wheel alone is

I = ½M(R²–r²) = ½7.7(0.7²–0.352) = 1.415 kg•m2

E = ½Iω² = ½I•9.8² = 67.9 J

For the cat initially

I = MR² = (7.7/4)0.7² = 0.943 kg•m²

E = ½I•9.8² = 45.3 J

we need angular momentum as that is conserved when the cat moves, and need that to calculate new ω

L = Iω = 9.8(1.415 + 0.943) = 23.11 kg•m²/s

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now cat moves.

cat new I = MR² = (7.7/4)0.35² = 0.236 kg•m²

ring's I is unchanged, so total I = 0.236 + 1.415 = 1.651 kg•m²

L = Iω

new ω = L/I = 23.11 / 1.651 = 14.00 rad/s

new E = ½Iω² = ½(1.651)14² = 160.8 J

ΔE = (160.8 – 45.3 )J =115.5 J

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Angular momentum in kg•m²/s

L = Iω

I is moment of inertia in kg•m²

ω is angular velocity in radians/sec

Angular kinetic energy E in Joules

E = ½Iω²

ω is angular velocity in radians/sec

I = moment of inertia in kg•m²

I is moment of inertia in kg•m²

I = cMR²

M is mass (kg), R is radius (meters)

c = 1 for a ring or hollow cylinder

c = 2/5 solid sphere around a diameter

c = 7/5 solid sphere around a tangent

c = ⅔ hollow sphere around a diameter

c = ½ solid cylinder or disk around its center

c = 1/12 rod around its center, R = length

c = ⅓ for a rod around its end, R = length

c = 1 for a point mass M at a distance R from

the axis of rotation

c = 1/3 for a door, where r is the width

for a wheel with r inner radius and R outer radius

I = ½M(R²–r²)

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