Worksheet: Center of Mass of Laminas

In this worksheet, we will practice finding the position of the center of gravity (mass) of a lamina composed of standard shapes in 2D.

Q1:

Where does the center of gravity of a fine rod, 𝐴𝐡, of uniform density lie?

  • AAt point 𝐡
  • BAt the midpoint of 𝐴𝐡
  • CAt point 𝐴

Q2:

A uniform lamina is bounded by the parallelogram 𝐴𝐡𝐢𝐷. Where does its center of gravity lie?

  • AAt point 𝐴
  • BAt point 𝐡
  • CAt the intersection point of the parallelogram’s diagonals

Q3:

Where does the center of gravity of a uniform triangular lamina lie?

  • AAt the intersection of its diagonals
  • BAt the intersection of its altitudes
  • CAt the intersection of its medians

Q4:

A uniform rod 𝐴𝐡𝐢 of length 46 cm was bent at its midpoint 𝐡 then suspended freely from 𝐴. Given that 𝐡𝐢 is horizontal when the rod is hanging in its equilibrium position, determine the distance between the center of gravity of the rod and 𝐴.

  • A23 cm
  • B 2 3 √ 2 2 cm
  • C 2 3 √ 2 3 cm
  • D 2 3 √ 2 cm

Q5:

Where does the center of gravity of a uniform circular disk lie?

  • AOn the circumference
  • BAt the center of the circle
  • CAt the center of the radius

Q6:

A uniform rod 𝐴𝐢 of length of 36 cm was bent from point 𝐡, where 𝐴𝐡=365cm, and π‘šβˆ π΄π΅πΆ=90∘. The rod was then suspended freely from 𝐴. Find the tangent of the angle that 𝐡𝐢 makes to the horizontal.

  • A 1 6 9
  • B 5 8
  • C16
  • D 9 1 6

Q7:

Two uniform rods, 𝐴𝐡 and 𝐡𝐢, of lengths π‘₯ and 𝑦, respectively, are connected at 𝐡. When the system is suspended from 𝐴 and settles in its equilibrium position, 𝐡𝐢 is horizontal. Given that 𝑦=136π‘₯, determine π‘šβˆ π΄π΅πΆ rounding your answer to the nearest minute if necessary.

  • A 2 8 2 0 β€² ∘
  • B 6 1 4 0 β€² ∘
  • C 4 1 2 1 β€² ∘
  • D 4 5 ∘

Q8:

A uniform lamina of mass π‘š is in the form of a rectangle 𝐴𝐡𝐢𝐷 in which 𝐴𝐡=48cm and 𝐡𝐢=128cm. The corner 𝐴𝐡𝐸, where 𝐸 is the midpoint of 𝐴𝐷, was cut off. The resulting lamina 𝐡𝐢𝐷𝐸 was freely suspended from the vertex 𝐢. A weight was placed at point 𝐷 which caused 𝐡𝐢 to be inclined at 45∘ to the vertical. Find the mass of the weight placed at point 𝐷 expressing your answer in terms of π‘š.

  • A 6 π‘š
  • B 4 9 π‘š
  • C 9 4 π‘š
  • D 1 6 π‘š

Q9:

A uniform lamina is shaped as a parallelogram 𝐴𝐡𝐢𝐷 such that 𝐴𝐡=34cm, 𝐴𝐷=23cm, and π‘šβˆ π΅π΄π·=60∘. The lamina was suspended from the point 𝐸 on 𝐷𝐢 which causes 𝐴𝐡 to be horizontal when the lamina is hanging in its position of equilibrium. Find the length of 𝐸𝐷.

Q10:

A uniform square lamina 𝐴𝐡𝐢𝐷 has a side length of 48 cm. A circular hole of area 256 cm2 was drilled into the lamina. The center of the circular hole lies on the diagonal 𝐡𝐷 and divides it into a ratio of 5∢1 from 𝐡. The lamina was suspended freely from point 𝐴 until it reached a state of equilibrium in a vertical plane. Given that the angle of inclination of side 𝐴𝐡 to the vertical is πœƒ, determine tanπœƒ.

  • A 1 3 1 1
  • B1
  • C 4 4 6 5
  • D 1 1 1 3

Q11:

A uniform triangular lamina 𝐴𝐡𝐢 is right-angled at 𝐡, 𝐡𝐢=17cm, 𝐴𝐡=17cm, and 𝑋, π‘Œ, and 𝑍 are the midpoints of 𝐴𝐡, 𝐡𝐢, and 𝐢𝐴 respectively. Triangle πΆπ‘Œπ‘ was cut off and then affixed to the lamina above triangle π‘Œπ΅π‘‹. The body was freely suspended from point 𝐡. Find the tangent of the angle that 𝐡𝐢 makes to the vertical, tanπœƒ, when the body is hanging in its equilibrium position.

  • A t a n πœƒ = 5 8
  • B t a n πœƒ = 7 8
  • C t a n πœƒ = 8 7
  • D t a n πœƒ = 8 5

Q12:

Find the position of the center of mass of the uniform lamina 𝐴𝐡𝐢, which is in the shape of an equilateral triangle.

  • A ο€Ώ 7 π‘Ž 2 , 7 √ 3 π‘Ž 2 
  • B ο€Ώ 1 4 π‘Ž 3 , 7 √ 3 π‘Ž 6 
  • C ο€Ώ 7 π‘Ž 3 , 7 √ 3 π‘Ž 6 
  • D ο€Ώ 7 π‘Ž 2 , 7 √ 3 π‘Ž 3 
  • E ο€Ώ 7 π‘Ž 2 , 7 √ 3 π‘Ž 6 

Q13:

A uniform triangular lamina has vertices 𝐴(7,1), 𝐡(9,3), and 𝐢(8,5). Find the coordinates of its center of mass.

  • A ( 8 , 3 )
  • B ( 2 4 , 9 )
  • C ( 8 , 9 )
  • D ( 2 4 , 3 )
  • E ( 5 , 1 )

Q14:

Two uniform lamina made of the same material are joined together to make one body. The first is a rectangle 𝐴𝐡𝐢𝐷 where 𝐴𝐡=16cm and 𝐡𝐢=7cm, and the second is an isosceles triangle 𝐢𝐸𝐷 where 𝐷𝐸=𝐢𝐸=17cm and vertex 𝐸 lies outside of the rectangle. Find the coordinates of the center of gravity of the lamina, given that rectangle 𝐴𝐡𝐢𝐷 is in the first quadrant, 𝐡 is at the origin, and 𝐢 is on the π‘₯-axis.

  • A ο€Ό 6 2 7 , 1 1 6 7 
  • B ο€Ό 2 2 9 1 4 , 1 1 6 7 
  • C ο€Ό 1 2 4 2 9 , 8 
  • D ο€Ό 2 2 9 2 9 , 8 

Q15:

The uniform lamina 𝐴𝐡𝐢𝐷 is a rectangle where 𝐴𝐡=48cm, 𝐡𝐢=64cm, and 𝐸∈𝐴𝐷 such that 𝐴𝐸=48cm. The corner 𝐴𝐡𝐸 is folded over along the line 𝐡𝐸 such that the side 𝐴𝐡 meets the side 𝐡𝐢 as shown in the figure. Find the coordinates of the center of mass of the lamina in this new shape.

  • A ( 2 0 , 1 2 )
  • B ( 8 , 1 8 )
  • C ( 2 6 , 1 8 )
  • D ( 1 4 , 1 8 )

Q16:

The given figure shows a uniform lamina bounded by a square of side length 4 cm. It is divided into nine congruent squares. If square 𝐸 is cut off, find the coordinates of the centre of gravity of the remaining part.

  • A ο€Ό 2 , 2 5 1 2 
  • B ο€Ό 2 5 1 2 , 2 5 1 2 
  • C ο€Ό 2 5 1 2 , 2 
  • D ( 4 , 2 )
  • E ( 2 , 2 )

Q17:

A uniform lamina in the form of an equilateral triangle of side length 24 cm has a mass of 298 g. A mass of 149 g is attached to the lamina at one of the trisection points of 𝐴𝐡 as shown in the figure. Determine the coordinates of the system’s center of gravity.

  • A ο€Ώ 3 2 3 , 8 √ 3 3 
  • B ο€Ώ 1 2 , 1 6 √ 3 3 
  • C ο€Ώ 3 2 3 , 1 6 √ 3 3 
  • D ο€» 1 2 , 4 √ 3 

Q18:

A uniform lamina in the form of a square 𝐴𝐡𝐢𝐷 of side length 28 cm has a mass of 54 grams. Masses of 10, 8, 4, and 8 grams are fixed at 𝐴, 𝐡, 𝐢, and 𝐷 respectively. Find coordinates of the center of mass of the system.

  • A ο€Ό 2 5 9 1 9 , 2 7 3 1 9 
  • B ( 1 3 , 1 3 )
  • C ( 2 2 , 1 3 )
  • D ο€Ό 1 3 , 6 2 3 

Q19:

A uniform rectangular lamina has a length of 63 cm and a width of 59 cm. It is divided into three equal sized rectangles along its length, the last of these rectangles has been folded over so that it lies flat on the middle rectangle as show in the figure. Find the coordinates of the centre of gravity of the lamina in this form.

  • A ο€Ό 4 9 2 , 5 9 2 
  • B ο€Ό 1 4 , 5 9 3 
  • C ο€Ό 5 9 3 , 1 4 
  • D ο€Ό 5 9 2 , 4 9 2 

Q20:

A uniform square lamina 𝐴𝐡𝐢𝐷 has a side length 𝑙. Another uniform lamina 𝐡𝐢𝐸 of the same density, shaped as an isosceles triangle, is attached to the square such that 𝐸 lies outside the square and 𝐡𝐸=𝐢𝐸. Given that the square’s side length is 53 times the length of the triangle’s height, find the center of mass of the system.

  • A ο€Ό 4 3 6 5 𝑙 , 1 2 𝑙 
  • B ο€Ό 2 8 6 5 𝑙 , 1 2 𝑙 
  • C ο€Ό 2 0 3 9 𝑙 , 1 3 𝑙 
  • D ο€Ό 4 6 6 5 𝑙 , 1 3 𝑙 

Q21:

A uniform lamina 𝐴𝐡𝐢𝐷 of mass 41 grams is shaped as a rhombus, where 𝐢𝐴=10cm and 𝐡𝐷=29cm. Masses of 36 g, 15 g, 18 g, and 30 g are fixed at the midpoints of 𝐴𝐡, 𝐡𝐢, 𝐢𝐷, and 𝐷𝐴 respectively. Find the distance between the center of gravity of the lamina and 𝐡𝐷.

  • A 5 6 cm
  • B 3 5 6 cm
  • C 9 4 cm
  • D 3 3 5 6 cm

Q22:

A uniform lamina is shaped as an equilateral triangle of side length 45 cm. What is the distance between the center of gravity and one of the vertices of the triangle?

  • A 4 5 2 cm
  • B 1 5 √ 3 cm
  • C 1 5 √ 3 2 cm
  • D 4 5 √ 3 2 cm

Q23:

The figure shows a uniform lamina that is symmetric about 𝐢𝐷. Given that 𝑙 is the distance from 𝐴𝐡 to the centre of gravity of the lamina, which of the following is true?

  • A 3 . 9 < 𝑙 < 6 c m cm
  • B 6 < 𝑙 < 6 . 8 c m cm
  • C 𝑙 = 3 . 9 c m
  • D 𝑙 = 6 c m

Q24:

Find the position of the center of mass of the uniform lamina 𝐴𝐡𝐢, which is in the shape of an equilateral triangle.

  • A ο€Ώ 1 5 π‘Ž 2 , 1 5 √ 3 π‘Ž 2 
  • B ο€Ό 1 5 π‘Ž 2 , 5 √ 3 π‘Ž 
  • C ο€Ώ 1 5 π‘Ž 2 , 5 √ 3 π‘Ž 2 
  • D ο€Ώ 5 π‘Ž , 5 √ 3 π‘Ž 2 
  • E ο€Ώ 1 0 π‘Ž , 5 √ 3 π‘Ž 2 

Q25:

The diagram shows a uniform plane figure. Given that the grid is composed of unit squares, find the coordinates of the figure’s center of mass.

  • A ο€Ό 4 2 πœ‹ + 1 4 3 6 πœ‹ + 3 0 , 8 3 
  • B ο€Ό 4 2 πœ‹ + 1 5 1 6 πœ‹ + 3 0 , 3 
  • C ο€Ό 4 2 πœ‹ + 1 4 3 6 πœ‹ + 3 0 , 3 
  • D ο€Ό 4 2 πœ‹ + 1 4 3 6 πœ‹ + 3 0 , 4 
  • E ο€Ό 4 2 πœ‹ + 1 5 1 6 πœ‹ + 3 0 , 8 3 

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