A thin uniform sheet of metal of uniform thickness is cut into the shape bounded by the line x = a and y = ± k x 2 , as shown. Find the coordinates of the centre of mass.

Both the upper part and lower part of the sheet are symmetric about x - axis . So, the centre of mass of given sheet lie along x - axis and hence y cm = 0 . Consider the surface density of the sheet is σ kg m - 2 . We know x cm = ∫ x d m ∫ d m . . . ( 1 ) Taking small strip of width d x at a distance x from origin.The area of the strip can be written as d A = 2 y d x We know σ = M A where M is mass and A is area. So, The Mass of the small strip d m = 2 σ y d x . Putting this value in equation 1 . We get x c m = ∫ x 2 σ y d x ∫ 2 σ y d x Putting y = k x 2 in above equation . We get x c m = ∫ x 2 σ k x 2 d x ∫ 2 σ k x 2 d x The limit is varying from x = 0 to x = a . Putting limit in above equation. we get x c m = ∫ 0 a k x 3 d x ∫ 0 a k x 2 d x x c m = a 4 4 a 3 3 = 3 a 4

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Five homogeneous bricks each of length $L$ are arranged as shown in the figure. Each brick is displaced with respect to the one in contact by $L / 5 .$ Find the $x$ -coordinate of the centre of mass relative to the origin $O$ shown.

Important Questions on Centre of Mass, Momentum and Collisions

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 5

A uniform disc of the radius

R

is put over another uniform disc of radius

2 R

made of the same material having the same thickness. The peripheries of the two discs touch each other. Locate the centre of mass of the system taking the centre of the large disc at the origin.

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 6

A thin uniform sheet of metal of uniform thickness is cut into the shape bounded by the line $x = a$ and $y = \pm k x^{2},$ as shown. Find the coordinates of the centre of mass.

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 7

A disc of radius R is cut out from a larger uniform disc of radius

2 R

in such a way that the edge of the hole touches the edge of the disc. Locate the centre of mass of remaining part.

HARD

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 8

Find the centre of mass of an annular half-disc shown in the figure.

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 9

Calculate the velocity of the centre of mass of the system of particles shown in the figure.

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 10

Two blocks of masses $10 kg$ and $30 kg$ are placed along a vertical line. The first block is raised through a height of $7 cm$ . By what distance should the second mass be moved to raise the centre of mass by $1 cm$ .

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 11

A projectile is fired from a gun at an angle of

45 °

with the horizontal and with a speed of

20 m s^{- 1}

relative to the ground. At the highest point in its flight, the projectile explodes into two fragments of equal mass. One fragment comes at rest just after the explosion. How far from the gun does the other fragment land, assuming a horizontal ground? Take

g = 10 m s^{- 2}

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Alpha Question Bank for Engineering: Physics>Chapter 9 - Centre of Mass, Momentum and Collisions>Exercise - 1>Q 12

A mercury thermometer is placed in a gravity-free hall without touching anything. As temperature rises mercury expands and ascends in the thermometer. If height ascended by mercury in the thermometer is

h

then by what height centre of mass of mercury and thermometer system descend?

Five homogeneous bricks each of length L are arranged as shown in the figure. Each brick is displaced with respect to the one in contact by L/5. Find the x-coordinate of the centre of mass relative to the origin O shown.

Important Questions on Centre of Mass, Momentum and Collisions

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Five homogeneous bricks each of length $L$ are arranged as shown in the figure. Each brick is displaced with respect to the one in contact by $L / 5 .$ Find the $x$ -coordinate of the centre of mass relative to the origin $O$ shown.