In the following diagram, determine the unknown mass M2(g). The center of mass o

Question

In the following diagram, determine the unknown mass M2(g). The center of mass of the meter stick is at 50.0 cm. Point A is at the 12.3 cm point on the meter stick and point B = 86.1 cm. The known mass is 100 g. In the above diagram, it was found that if the unknown weight is removed, the fulcrum C moves 14 cm to the left. Determine the mass of the meter stick using the equation WiXi = WX In the above diagram, determine the position of W1 if the W2 is moved to 5 cm to the left, assuming that the fulcrum is at the center of mass of the meter stick.

Explanation / Answer

(a)

r2 = CB = 86.1-50 = 36.1 cm
r1 = CA = 50-12.3 = 37.7 cm

M1 = 100 g = 0.1 Kg and the meter stick is balanced on its center point and half its mass M0 on each side, with center of mass distances on each side r0 = 25.0 cm. Therefore, for equilibrium we must have:

r0M0/2 + r1 M1 = r0M0/2 + r2 M2 => r1 M1 = r2 M2

So (* is multiplication) : 0.1 * 37.7 = 36.1 M2 =>

M2 = 3.77 / 36.1 = 0.1044321329639889... or approx 104.4g

(b)

If we remove M2 and the center of equilibrium is now 14 cm to the left we have:

r1 = CA = (50-14) - 12.3 = 36 - 12.3 =23.7 cm

M1 = 100 g = 0.1 Kg and the meter stick is balanced on its new center point its mass M0 split on the left (L) and right (R) as follows:

M0L = 36.M0/100 (Kg), with its center of mass at r0L = 36/2 = 18cm (0.18 m) to the left of the fulcrum, and

M0R = 64.M0/100 (Kg), with its center of mass at r0R = 64/2 = 32cm (0.32 m) to the right of the fulcrum, We Therefore, for equilibrium we must have:

r0LM0L + r1 M1 = r0RM0R => r1 M1 = r0RM0R - r0LM0L = r0RM0R - r0LM0L

And  (* is multiplication) => 0.237 * 0.1 = ( 0.32 * 0.64 - 0.18 * 0.36 ) M0

Or: M0 = (0.237 * 0.1) / ( 0.32 * 0.64 - 0.18 * 0.36 ) = 0.0237 / 0.14 = 0.1692857142... Kg or approximately 169.3 g

(c) We are back to case (a) with modified distances.

r2 = CB = 36.1 cm - 5 cm = 31.1 cm or 0.311 m
r1 = CA = now unknown.

M1 = 100 g = 0.1 Kg

M2 = 104.432133 g = 0.104432133 Kg

r0R = 55/2 cm = 0.275 m

r0L = 45/2 = 0.225 m

M0L =  0.1692857142 * 0.45 = 0.076178571428571428571428571428571

M0R = 0.1692857142 * 0.55 = 0.093107142857142857142857142857143

So: r0LM0L + r1 M1 = r0RM0R + r2 M2 => r1 = ( r0RM0R - r0LM0L + r2 M2 ) / M1

Hence r1 = (0.275 * 0.09311 - 0.225 * 0.07618 + 0.311 * 0.104432) / 0.1 = 0.409431 m or approx 41 cm

THE LOGIC IS GOOD BUT CHECK THE CALCULATIONS PLEASE.

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