Worksheet: Moment of Inertia of Solids

In this worksheet, we will practice finding the moment of inertia and radius of gyration of a solid and using the parallel axis theorem to find the moment of inertia of a composite solid at different axes.

Q1:

Matthew has a model train that uses a circular cone as a flywheel. The cone has radius π‘Ÿ and height β„Ž.

Find its moment of inertia about its axis of symmetry.

  • A 2 π‘š β„Ž π‘Ÿ 5 
  • B 2 π‘š π‘Ÿ 3 
  • C 3 π‘š β„Ž π‘Ÿ 1 0 
  • D π‘š π‘Ÿ 
  • E 3 π‘š π‘Ÿ 1 0 

What is its radius of gyration about this axis?

  • A π‘Ÿ
  • B ο„ž 3 2 π‘Ÿ β„Ž
  • C ο„ž 3 1 0 π‘Ÿ
  • D 3 1 0 π‘Ÿ
  • E π‘Ÿ β„Ž

Matthew would like to replace the cone with a circular cylinder of the same mass. He does not want this change to affect the performance of his train, so he needs a cylinder with the same moment of inertia as the cone. Find an expression for the radius 𝑅 of the cylinder that he needs to buy.

  • A 𝑅 = 3 1 0 π‘Ÿ
  • B 𝑅 = ο„ž 3 5 π‘Ÿ
  • C 𝑅 = 1 2 ο„ž 3 5 π‘Ÿ
  • D 𝑅 = ο„ž 3 1 0 π‘Ÿ
  • E 𝑅 = 3 5 π‘Ÿ

Q2:

Suppose 𝑛 particles are attached to a light circular ring with radius π‘Ÿ. If the total mass of the particles is 𝑀, find the moment of inertia of the resulting system about an axis passing through the center of the ring which is perpendicular to the plane of the ring.

  • A 4 𝑀 π‘Ÿ 
  • B 𝑀 π‘Ÿ 
  • C 𝑀 π‘Ÿ
  • D 2 𝑀 π‘Ÿ 
  • E 𝑀 π‘Ÿ 𝑛 

Q3:

Jennifer is building a model race car. She wants the car to be as responsive as possible, so she is looking for a drive wheel with the smallest moment of inertia. The figure shows the configuration of the drive wheel, π·π‘Š, in relation to the engine’s output shaft, 𝑂, and the two drive shafts, π·π‘†οŠ§ and π·π‘†οŠ¨.

She has two options for the drive wheel. The specifications for each drive wheel are detailed in the table.

What is the moment of inertia of drive wheel A about a perpendicular axis passing through the center of the disk? Give your answer in gram square millimeters.

What is the moment of inertia of drive wheel B about a perpendicular axis passing through the center of the disk? Give your answer in gram square millimeters.

Which drive wheel should she choose to maximize the responsiveness of the model car?

  • ADrive wheel A
  • BDrive wheel B
  • CEither one

Q4:

Three light rods, each of length 2π‘Ž, are joined together to form a triangle 𝐴𝐡𝐢. Three particles of masses π‘š, π‘š, and 2π‘š are attached to the vertices 𝐴, 𝐡, and 𝐢 respectively. Find the moment of inertia of the resulting system about an axis passing through the vertex 𝐴 which is parallel to the base 𝐡𝐢.

  • A 1 0 π‘š π‘Ž 
  • B 3 √ 3 π‘š π‘Ž
  • C 6 π‘š π‘Ž 
  • D 9 π‘š π‘Ž 
  • E 9 π‘š π‘Ž

Q5:

Three uniform rods, each of length 2π‘Ž and mass 𝑀, are rigidly joined at their ends to form an equilateral triangle 𝐴𝐡𝐢. Find the moment of inertia of this triangular structure about an axis passing through the vertex 𝐴 which is parallel to the base 𝐡𝐢.

  • A 1 1 3 𝑀 π‘Ž 
  • B 1 4 𝑀 π‘Ž 
  • C 5 𝑀 π‘Ž 
  • D 3 𝑀 π‘Ž 
  • E 1 7 3 𝑀 π‘Ž 

Q6:

The moment of inertia of a thin uniform rod of mass π‘š and length 2π‘Ž about an axis which is perpendicular to the rod and passes through its center is π‘šπ‘Ž3. Using the parallel axes theorem, find the moment of inertia of the rod about an axis which is perpendicular to the rod and passes through one of its ends.

  • A 5 π‘š π‘Ž 6 
  • B 7 π‘š π‘Ž 1 2 
  • C 4 π‘š π‘Ž 3 
  • D 3 π‘š π‘Ž 4 
  • E π‘š π‘Ž 3 

Q7:

Four uniform rods, each of mass 𝑀 and length π‘Ž, are rigidly joined at their ends to form a square. Find the moment of inertia of this square about one of its diagonals. You may use the fact that the moment of inertia of a rod of length 𝑙 and mass π‘š about an axis passing through one end and inclined at an angle of πœƒ to the rod is 13π‘šπ‘™πœƒοŠ¨οŠ¨sin.

  • A 4 𝑀 π‘Ž 
  • B 𝑀 π‘Ž 
  • C 2 3 𝑀 π‘Ž 
  • D 1 6 𝑀 π‘Ž 
  • E 1 3 𝑀 π‘Ž 

Q8:

Find the moment of inertia of a uniform solid sphere of mass π‘š and radius π‘Ž about an axis which is tangent to a point on its surface.

  • A 5 π‘š π‘Ž 2 
  • B 2 π‘š π‘Ž 5 
  • C 3 π‘š π‘Ž 2 
  • D 7 π‘š π‘Ž 5 
  • E π‘š π‘Ž 2 

Q9:

A uniform semicircular lamina has mass π‘š and radius π‘Ž. Find the moment of inertia of the lamina about an axis which is perpendicular to the lamina and passes through its center of mass.

  • A π‘š π‘Ž 4 
  • B π‘š π‘Ž 2 
  • C π‘š π‘Ž 1 8 πœ‹ ο€Ή 9 πœ‹ βˆ’ 3 2    
  • D π‘š π‘Ž 1 8 πœ‹ ο€Ή 3 πœ‹ βˆ’ 1 6    
  • E π‘š π‘Ž 9 πœ‹ ο€Ή 9 πœ‹ βˆ’ 1 6    

Q10:

A uniform annulus of mass π‘š is made from a disc of radius 𝑅 by cutting out a concentric disc of radius π‘Ÿ. Find the moment of inertia of the annulus about an axis which passes through its centre and is perpendicular to the plane of the annulus. You may use the fact that the moment of inertia of a solid disc of radius π‘Ž and mass π‘š about an axis which passes through its centre and is perpendicular to the plane of the disc is 12π‘šπ‘ŽοŠ¨.

  • A π‘š ο€Ή 𝑅 + π‘Ÿ  2 ( 𝑅 βˆ’ π‘Ÿ ) οŠͺ οŠͺ  
  • B 1 2 π‘š ο€Ή 𝑅 + π‘Ÿ   
  • C 1 2 π‘š π‘Ÿ 
  • D 1 2 π‘š 𝑅 
  • E 1 2 π‘š ο€Ή 𝑅 βˆ’ π‘Ÿ   

Q11:

The moment of inertia of a uniform circular disc of mass π‘š and radius π‘Ÿ about an axis which passes through its centre and is perpendicular to its plane is π‘šπ‘Ÿ2. Use the parallel axes theorem to find the moment of inertia of the disc about an axis which is perpendicular to the plane of the disc and passes through a point on its circumference.

  • A 3 π‘š π‘Ÿ 2 
  • B π‘š π‘Ÿ 2 
  • C π‘š π‘Ÿ 
  • D 3 π‘š π‘Ÿ 4 
  • E 5 π‘š π‘Ÿ 4 

Q12:

A uniform wire of length 3π‘Ž and mass π‘š is bent into the shape of an equilateral triangle. Find the moment of inertia of the triangle about an axis perpendicular to the triangle which passes through one of its vertices.

  • A 3 π‘š π‘Ž 2 
  • B π‘š π‘Ž 4 
  • C π‘š π‘Ž 
  • D 7 π‘š π‘Ž 4 
  • E 3 π‘š π‘Ž 4 

Q13:

One end of a uniform rod of mass 𝑀 and length 𝑙 is attached to the center of a uniform disk of radius π‘Ÿ so that the rod is perpendicular to the plane of the disk. Find the moment of inertia of the system about an axis along the rod. You may use the fact that the moment of inertia of a solid disk of radius 𝑑 and mass π‘š about an axis which passes perpendicularly through the center of the disk is 12π‘šπ‘‘οŠ¨.

  • A 𝑀 𝑙 
  • B 1 2 π‘š π‘Ÿ βˆ’ 𝑀 𝑙  
  • C 𝑀 𝑙 βˆ’ 1 2 π‘š π‘Ÿ  
  • D 1 2 π‘š π‘Ÿ 
  • E 1 2 π‘š π‘Ÿ + 𝑀 𝑙  

Q14:

A uniform ring of radius π‘Ÿ and mass π‘š has a particle of mass π‘š attached to it. Find the moment of inertia of the composite body about an axis passing through the center of the ring and perpendicular to the plane of the ring.

  • A 2 π‘š π‘Ÿ
  • B π‘š π‘Ÿ
  • C 2 π‘š π‘Ÿ  
  • D 2 π‘š π‘Ÿ 
  • E π‘š π‘Ÿ 

Q15:

Find the moment of inertia of a uniform solid sphere of radius π‘Ž and mass π‘š about an axis which lies at a distance √2π‘Ž2 from the center of the sphere.

  • A 1 3 π‘š π‘Ž 2 0 
  • B π‘š π‘Ž 2 
  • C 9 π‘š π‘Ž 1 0 
  • D 2 π‘š π‘Ž 5 
  • E 7 π‘š π‘Ž 5 

Q16:

Find the moment of inertia of a thin uniform rod of mass π‘š and length 2𝑙 about an axis parallel to the rod and at a distance 𝑑 from it.

  • A π‘š 𝑑 
  • B 1 2 π‘š 𝑑 
  • C π‘š 𝑑 
  • D π‘š 𝑑
  • E π‘š 𝑑  

Q17:

A uniform rod of mass 2π‘š and length 𝑙 has a particle of mass π‘š fixed to one end. Find the moment of inertia of the system about an axis which passes through the other end of the rod and is perpendicular to it. You may use the fact that the moment of inertia of a rod of length 𝑙 and mass π‘š about an axis passing through the end of the rod and perpendicular to it is 13π‘šπ‘™οŠ¨.

  • A 1 3 π‘š 𝑙 
  • B 5 3 π‘š 𝑙 
  • C π‘š 𝑙 
  • D 4 3 π‘š 𝑙 
  • E 2 π‘š 𝑙 

Q18:

Three light rods are joined together to form the right triangle 𝐴𝐡𝐢, as shown in the diagram. Three particles of masses π‘š, 2π‘š, and 3π‘š are attached to the vertices 𝐴, 𝐡, and 𝐢 respectively. Find the moment of inertia of the resulting system about an axis passing through the vertex 𝐴 which is parallel to the base 𝐡𝐢.

  • A 1 5 π‘š π‘Ž
  • B 4 5 π‘š π‘Ž
  • C 2 7 π‘š π‘Ž 
  • D 4 5 π‘š π‘Ž 
  • E 4 6 π‘š π‘Ž 

Q19:

Three light rods are joined together to form the isosceles triangle 𝐴𝐡𝐢, as shown in the diagram. Three particles of masses π‘š, π‘š, and 2π‘š are attached to the vertices 𝐴, 𝐡, and 𝐢 respectively. Find the moment of inertia of the resulting system about an axis passing through the vertex 𝐴 which is parallel to the base 𝐡𝐢.

  • A 2 7 π‘š π‘Ž
  • B 1 8 π‘š π‘Ž 
  • C 2 7 π‘š π‘Ž 
  • D 9 π‘š π‘Ž
  • E 2 8 π‘š π‘Ž 

Q20:

Find the moment of inertia of a uniform circular disk of mass π‘š and radius π‘Ÿ about the axis perpendicular to it which passes through its center.

  • A 1 2 π‘š π‘Ÿ 
  • B π‘š π‘Ÿ 
  • C 8 π‘š π‘Ÿ 
  • D 2 3 π‘š π‘Ÿ
  • E 1 4 π‘š π‘Ÿ 

Q21:

Find the moment of inertia of a thin uniform rod of mass π‘š and length 2𝑙 about an axis which is perpendicular to the rod and passes through its center.

  • A 1 1 2 π‘š 𝑙 
  • B 1 6 π‘š 𝑙 
  • C 1 3 π‘š 𝑙 
  • D 1 6 π‘š 𝑙
  • E 1 2 π‘š 𝑙

Q22:

Find the moment of inertia of a thin uniform rod of mass π‘š and length 2𝑙 about an axis inclined at an angle of πœƒ to the rod which passes through one of its ends.

  • A π‘š 𝑙 πœƒ s i n
  • B 4 π‘š 𝑙 πœƒ 3   c o s
  • C 4 π‘š 𝑙 πœƒ 3   s i n
  • D π‘š 𝑙 πœƒ 3   s i n
  • E 4 π‘š 𝑙 3 

Q23:

Find the moment of inertia of a thin uniform rod of mass π‘š and length 𝑙 about an axis inclined at an angle of πœ‹3 to it which passes through its center.

  • A 1 4 8 π‘š 𝑙 
  • B 1 3 2 π‘š 𝑙 
  • C 3 1 6 π‘š 𝑙 
  • D 1 1 6 π‘š 𝑙 
  • E 1 4 π‘š 𝑙 

Q24:

Find the moment of inertia of a thin uniform rod of mass π‘š and length 2𝑙 about an axis inclined at an angle of πœ‹6 to the rod which passes through the rod at a distance of 𝑙4 from its center.

  • A 1 9 1 9 2 π‘š 𝑙 
  • B 1 3 π‘š 𝑙 
  • C 1 9 6 4 π‘š 𝑙 
  • D 1 8 π‘š 𝑙
  • E 1 9 9 6 π‘š 𝑙 

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