This set of Physics Multiple Choice Questions & Answers (MCQs) focuses on “Theorems of Perpendicular and Parallel Axes”.

1. A planar body is lying in the xz plane. What is the relation between its moment of inertia along the x, y & z axes?

a) I_{z} = I_{x} + I_{y}

b) I_{x} = I_{x} + I_{z}

c) I_{y} = I_{x} + I_{z}

d) I_{z} = I_{x} = I_{y}, because body is planar

View Answer

Explanation: The body is lying in the xz plane. The perpendicular axis theorem states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of moment of inertia about two perpendicular axes in the plane of the body. I

_{y}is perpendicular to plane of body, so I

_{y}= I

_{x}+ I

_{z}.

2. Perpendicular axis theorem can be applied for which of the following bodies?

a) Ring having radius R & negligible cross section

b) Disc of radius R and thickness t

c) Cylinder of radius R and height h

d) A cube of side ‘a’

View Answer

Explanation: Perpendicular axis theorem can only be applied to planar bodies. From the given options only the ring is planar.

3. Consider two perpendicular axis in the plane of a planar body, such that I_{1} = 2 I_{2}. The moment of inertia about an axis perpendicular to the plane and passing through intersection of I_{1} & I_{2} is 9kgm^{2}. Find the value of I_{1}& I_{2}.

a) I_{1} = 9kg m^{2}, I_{2} = 4.5kgm^{2}

b) I_{1} = 3kg m^{2}, I_{2} = 6kg m^{2}

c) I_{1} = 6kg m^{2}, I_{2} = 3kg m^{2}

d) I_{1} = 18kg m^{2}, I_{2} = 9kg m^{2}

View Answer

Explanation: By perpendicular axis theorem: 9 = I

_{1}+ I

_{2}= 3 I

_{2}.

∴ I

_{2}= 3kgm

^{2}& I

_{1}= 2*3 = 6kgm

^{2}.

4. The moment of inertia of a planar disc about a diameter is 8kgm^{2}. What is the moment of inertia about an axis passing through its centre and perpendicular to the plane of disc?

a) 8kgm^{2}

b) 16kgm^{2}

c) 4kgm^{2}

d) 2√2kgm^{2}

View Answer

Explanation: The moment of inertia about any diametrical axis will be the same. So, we can consider two perpendicular diameters and use the perpendicular axis theorem to get the moment of inertia about the axis which is perpendicular to the plane.

Thus, moment of inertia = 8+8

= 16kgm

^{2}.

5. Let I_{1} be the moment of inertia about the centre of mass of a thick asymmetrical body. Let I_{2} be the moment of inertia about an axis parallel to I_{1}. The distance between the two axes is ‘a’ & the mass of the body is ‘m’. Find the relation between I_{1} & I_{2}.

a) I_{2} = I_{1} – ma^{2}

b) I_{1} = I_{2} – ma^{2}

c) I_{2} = I_{1}

d) Parallel axis theorem can’t be used for a thick asymmetrical body

View Answer

Explanation: Parallel axis theorem can be used for any body. The parallel axis theorem states that moment of inertia about an axis perpendicular to an axis passing through centre of mass is given by:

I = I

_{COM}+ ma

^{2}, where m is mass of the body & ‘a’ is the distance between the axes. So, I

_{2}= I

_{1}+ ma

^{2}OR I

_{1}= I

_{2}– ma

^{2}.

6. What is the moment of inertia of a rod, of mass 1kg & length 6m, about an axis perpendicular to rod’s length and at a distance of 1.5m from one end?

a) 0.75kgm^{2}

b) 3kgm^{2}

c) 5.25kgm^{2}

d) 14.25kgm^{2}

View Answer

Explanation: Moment of inertia about an axis perpendicular to length and passing through COM is equal to MI

^{2}/12. To find I about given axis, we use parallel axis theorem,

I = I

_{COM}+ ma

^{2}, where a is the distance between the axis and m is mass of the body.

I = MI

^{2}/12 + Ma

^{2}

= 1*36/12 + 1*2.25

= 3 + 2.25 = 5.25kgm

^{2}.

7. The moment of inertia of a ring about a tangent is 4kgm^{2}. What is the moment of inertia about an axis passing through the centre of the ring and perpendicular to its plane? Mass of the ring is 2kg & diameter is 2m.

a) 2kgm^{2}

b) 4kgm^{2}

c) 8kgm^{2}

d) 1kgm^{2}

View Answer

Explanation: Using parallel axis theorem we can find I about centre of mass.

∴ I = 4 – 2*1 = 2kgm

^{2}.

Now using perpendicular axis theorem we get I

_{1}about the desired axis.

∴ I

_{1}= I + I = 4kgm

^{2}.

8. The moment of inertia of a planar square about a planar axis parallel to one side is 10kgm^{2}. What is the moment of inertia about a diagonal?

a) 10kgm^{2}

b) 5kgm^{2}

c) 20kgm^{2}

d) 1kgm^{2}

View Answer

Explanation: Using perpendicular axis theorem we can say that sum of moment of inertia about two perpendicular axes will be the same as sum of moment of inertia about other 2 perpendicular axes. Let I

_{1}be the moment of inertia about a diagonal, we can say:

I

_{1}+ I

_{1}= 10 + 10 = 20kgm

^{2}

Or I

_{1}=10kgm

^{2}.

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