A barbell is 1.5 m long. Three weights, each of mass 20 kg, are hung on the left and two weights of the same mass, on the right. The width of each weight is 4 cm and each group of weights is placed 4 cm from the ends. Where is the center of mass of the barbell as measured from the mid-point, M, of the bar? The bar is of uniform mass and has mass 5 kg, and the retaining collars are of negligible mass. Take to the right as positive.
A) -5.90 cm
B) -5.62 cm
C) +5.62 cm
D) +5.90 cm
E) None of the above

Question

A barbell is 1.5 m long. Three weights, each of mass 20 kg, are hung on the left and two weights of the same mass, on the right. The width of each weight is 4 cm and each group of weights is placed 4 cm from the ends. Where is the center of mass of the barbell as measured from the mid-point, M, of the bar? The bar is of uniform mass and has mass 5 kg, and the retaining collars are of negligible mass. Take to the right as positive. 
A) -5.90 cm 
B) -5.62 cm 
C) +5.62 cm 
D) +5.90 cm 
E) None of the above

Quizplus · Accepted Answer

First, we need to calculate the total mass of the barbell.

Mass of one weight = 20 kg

Mass of three weights on the left = 20 x 3 = 60 kg

Mass of two weights on the right = 20 x 2 = 40 kg

Mass of the bar = 5 kg

Total mass = 60 + 40 + 5 = 105 kg

Now, let's find the center of mass.

We can assume that the barbell is a system of six point masses:

- Mass #1: weight on the left, 4 cm from the left end 
- Mass #2: weight on the left, 4 cm from the mid-point M 
- Mass #3: weight on the left, 4 cm from the mid-point M 
- Mass #4: weight on the right, 4 cm from the mid-point M 
- Mass #5: weight on the right, 4 cm from the right end 
- Mass #6: the bar itself

We need to find the position of the center of mass, x_cm, relative to the mid-point M.

Using the formula for the center of mass of a system of point masses:

x_cm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)

where mi is the mass of point mass i and xi is its position relative to some reference point (in this case, the mid-point M).

Let's take the mid-point M as our reference point, so the x's become the distances from the mid-point.

x1 = -1.5/2 + 0.04 = -0.71 m (left) 
m1 = 20 kg

x2 = -0.04 m (left) 
m2 = 20 kg

x3 = 0.04 m (right) 
m3 = 20 kg

x4 = 0.04 m (right) 
m4 = 20 kg

x5 = 1.5/2 - 0.04 = 0.71 m (right) 
m5 = 20 kg

x6 = 0 m (the bar) 
m6 = 5 kg

Plugging in the numbers:

x_cm = [(20)(-0.71) + (20)(-0.04) + (20)(0.04) + (20)(0.04) + (20)(0.71) + (5)(0)] / (105)

x_cm = -0.0476 m

Therefore, the center of mass is about 4.76 cm to the left of the mid-point M.

To find the position relative to the left end (which is what the answer choices are in), we need to add 0.04 m:

x_cm_left = -0.0476 + 0.04 = -0.0076 m

x_cm_left = -0.76 cm

Rounding to two decimal places, we get -5.62 cm as the answer, which is choice B.

A Barbell Is 1