Question

For each of the pairs of objects below, decide which object has the larger moment of inertia, selecting from the following answers: A) the first B) the second C) neither D) can't tell E.g., if the answer for the first pair is B and for the rest, D, enter BDDDDD. The objects are of uniform density. Unless otherwise indicated, the flat objects rotate about an axis perpendicular to the plane in which they lie.

1.(A) A thin circular sheet of material of radius a/sqrt(2) rotating about its centre, or (B) a thin square sheet of the same material of side a, also rotating about its centre.

2.(A) A thin square sheet of material of side a rotating about its centre, or (B) a thin sheet of the same material in the form of an equilateral triangle of side a, also rotating about its centre.

3.(A) A thin circular sheet of material of radius a/2 rotating about a tangent, or (B) a thin square sheet of the same material of side a, rotating about its lower edge.

4.(A) A rod of mass m and length ℓ rotating about its centre, or (B) a rod of mass m and length 2ℓ, also rotating about its centre.

5.(A) A sphere rotating about its axis, or (B) the same sphere rotating about a parallel axis, tangent to the surface.

6.(A) A cylindrical shell rolling on a table, or (B) a solid cylinder of identical mass and size, also rolling.

Answer 1

1.(A) A thin circular sheet of material of radius a/sqrt(2) rotating about its centre,

IA = (1/2)*M*R^2 = (1/2)*d*pi*R^2*R^2 = (1/2)*d*pi*R^4 = IA = (pi/8)*d*a^4 = 0.4*pi*a^4

(B) a thin square sheet of the same material of side a, also rotating about its centre.

IB = (1/6)*m*a^2 = (1/6)*d*a^2*a^2

IB = 0.17*d*a^4

A)

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2.

(A) A thin square sheet of material of side a rotating about its centre,

IA = (1/6)*d*a^2*a^2 = 0.17 *d*a^2

(B) a thin sheet of the same material in the form of an equilateral triangle of side a, also rotating about its centre.

IB = (1/12)*m*a^2

IB = (1/12)*d*(1/2)*a*a*sqrt3/2*a^2

IB = 0.04*d*a^2

A

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3.(A) A thin circular sheet of material of radius a/2 rotating about a tangent,

IA = (5/16)*d*a^4 = 0.3125 *d*a^2

(B) a thin square sheet of the same material of side a, rotating about its lower edge.

IB = 0.33*d*a^4

B

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4.(A) A rod of mass m and length ℓ rotating about its centre,

IA = (1/12)*m*l^2 = 0.083*m*l^2

(B) a rod of mass m and length 2ℓ, also rotating about its centre.

IB = (1/12)*m*(2l)^2 = (1/3)*m*l^2 = 0.33*m*l^2

B

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5.(A) A sphere rotating about its axis,

IA = (2/5)*m*R^2 = 0.4*m*R^2

(B) the same sphere rotating about a parallel axis, tangent to the surface.

IB = (2/5)*m*R^2 + m*R^2 = (7/5)*m*R^2 = 1.4 m*R^2

B

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6.(A) A cylindrical shell rolling on a table,

IA = (1/2)*M*(R1^2+R2^2)

(B) a solid cylinder of identical mass and size, also rolling

IB = (1/2)*M*R2^2

A
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