Question 14 of 16intermediate🔧 ApplyNumerical3 marks

From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

Correct Answer

The centre of gravity of the resulting flat body is located at a distance of R/6 from the centre of the original disk, on the side opposite to the hole.

Step 1: Mass calculation using area proportionality Let mass of original disk = M Area of original disk = πR² Area of hole = π(R/2)² = πR²/4 Since the disk is uniform, mass ∝ area Mass of hole = M/4 Mass of remaining body = M - M/4 = 3M/4

Step 2: Applying centre of mass formula Let the centre of original disk be origin O. The centre of hole is at distance R/2 from O. Let the centre of gravity of remaining body be at distance x from O (opposite to hole).

Using the principle: Centre of mass of original disk = weighted average of hole and remaining body

M × 0 = (M/4) × (R/2) + (3M/4) × x

Step 3: Solving for x 0 = MR/8 + 3Mx/4 3Mx/4 = -MR/8 x = -R/6

The negative sign indicates the centre of gravity lies on the opposite side of the hole from the centre of the original disk, at distance R/6 from it.

Exercise: EXERCISES | Q: 6.15 | (Page 35)
For More Understanding

Explanation

This question applies the concept that for a uniform body, the centre of mass coincides with the geometric centre. When a portion is removed, we treat it as a negative mass. The original disk's centre of mass is at its geometric centre (origin). Using the formula X = (1/M)∫x dm, we can find the new centre of mass by considering the remaining mass and its distribution. The hole's mass is subtracted from the original mass, and the calculation uses the weighted average principle.

Solution Steps

  1. Step 1: Calculate masses using area proportionality - original disk has mass M, hole has mass M/4, remaining body has mass 3M/4

  2. Step 2: Set up centre of mass equation treating the hole as negative mass at distance R/2 from origin

  3. Step 3: Solve M×0 = (M/4)×(R/2) + (3M/4)×x to get x = -R/6

  4. Step 4: Interpret the result - centre of gravity is R/6 from centre on side opposite to hole

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