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Worksheet: Center of Mass

Q1:

Find the center of mass of the region 𝑅 =  ( π‘₯ , 𝑦 ) 𝑦 β‰₯ 0 , π‘₯ + 𝑦 ≀ 1  : 2 2 with the given density function 𝜌 ( π‘₯ , 𝑦 ) = 𝑦 .

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

Q2:

Find the center of mass of the region 𝑅 =  ( π‘₯ , 𝑦 ) 0 ≀ π‘₯ ≀ 1 , 0 ≀ 𝑦 ≀ π‘₯  : 2 with the given density function 𝜌 ( π‘₯ , 𝑦 ) = π‘₯ + 𝑦 .

  • A ο€Ό 3 0 7 , 5 5 1 4 7 
  • B ο€Ό 5 5 1 4 7 , 1 7 2 1 
  • C ο€Ό 1 7 9 0 , 1 1 1 2 6 
  • D ο€Ό 1 7 2 1 , 5 5 1 4 7 
  • E ο€Ό 2 1 1 7 , 1 4 7 5 5 

Q3:

Find the center of mass of the region 𝑅 = { ( π‘₯ , 𝑦 ) 0 ≀ π‘₯ ≀ 2 , 0 ≀ 𝑦 ≀ 4 } : with the given density function 𝜌 ( π‘₯ , 𝑦 ) = 2 𝑦 .

  • A ο€Ό 2 , 1 6 3 
  • B ο€Ό 8 3 , 1 
  • C ο€Ό 1 6 3 , 2 
  • D ο€Ό 1 , 8 3 
  • E ο€Ό 1 , 3 8 

Q4:

Find the center of mass of the region 𝑅 =  ( π‘₯ , 𝑦 ) 𝑦 β‰₯ 0 , π‘₯ β‰₯ 0 , 1 ≀ π‘₯ + 𝑦 ≀ 4  : 2 2 with the given density function 𝜌 ( π‘₯ , 𝑦 ) = √ π‘₯ + 𝑦 2 2 .

  • A ο€Ό 0 , 4 5 πœ‹ 1 4 
  • B ο€Ό 4 5 πœ‹ 1 4 , 0 
  • C ο€Ό 7 πœ‹ 6 , 7 πœ‹ 6 
  • D ο€Ό 4 5 πœ‹ 1 4 , 4 5 πœ‹ 1 4 
  • E ο€Ό 0 , 7 πœ‹ 6 

Q5:

Find the center of mass of the region 𝑅 =  ( π‘₯ , 𝑦 ) 𝑦 β‰₯ 0 , π‘₯ + 𝑦 ≀ π‘Ž  : 2 2 2 with the given density function 𝜌 ( π‘₯ , 𝑦 ) = 1 .

  • A ο€Ό 4 π‘Ž 3 πœ‹ , 4 π‘Ž 3 πœ‹ 
  • B ο€Ό 4 π‘Ž 3 πœ‹ , 0 
  • C ο€Ό 2 π‘Ž 3 πœ‹ , 0 
  • D ο€Ό 0 , 4 π‘Ž 3 πœ‹ 
  • E ο€Ό 0 , 2 π‘Ž 3 πœ‹ 