A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of x and y coordinates of center of mass of plate in n/9. The value of n is ________.

Solution

Physics sir · 4 steps

Treat the plate as a full rectangle with a square cut out, and give the missing square negative area.

1. Break the shape. The plate is the full 3 × 2 rectangle (area 6, centre at (1.5, 1)) minus the 1 × 1 notch (area 1, centre at (1.5, 1.5)). The plate is uniform, so mass is proportional to area and the 10 kg never enters the ratio.
2. x coordinate. The shape is symmetric about the line x = 1.5, so x_(cm) = 1.5.

   y_(cm) = (6 × 1 - 1 × 1.5)/(6 - 1) = 4.5/5 = 0.9

3. Take the ratio. x_(cm)/y_(cm) = 1.5/0.9 = 5/3 = 15/9.
4. So. The ratio is n/9 with n = 15.

The common slip: 5.4. Taking the ratio as y to x gives 0.9/1.5 = 3/5, so n = 5.4, not 15.

The idea to keep. For a plate with a hole, find the centre of mass of the full shape and subtract the removed piece with negative area. Use symmetry first, because it can give one coordinate for free.

Answer from the NTA final answer key. Solution by PrepWiser.