Two Falling Stones and Their Center of Mass

In summary, two stones are dropped from the same point, one at t = 0 and the other at t = 250 ms with twice the mass. At t = 600 ms, the center of mass of the two-stone system is 0.012 meters below the release point and is moving with a velocity determined by solving the equation for the position of the center of mass and taking the first derivative.
  • #1
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A stone is dropped at t = 0. A second stone, with twice the mass of the first, is dropped from the same point at t = 250 ms.
(a) How far below the release point is the center of mass of the two stones at t = 600 ms? (Neither stone has yet reached the ground.)
(b) How fast is the center of mass of the two-stone system moving at that time?

First and foremost: the times are 0.250s and 0.600s respectively.
I set my co-ordinate system with x=0 at the release point and the x-axis is positive in the downward direction (sorry).

Other than that...

I first found (or perhaps tried to find) the center of mass at 0.250s and said that the COM moves with a constant acceleration of 9.81m/s^2.

PART A:
To find where the first stone was at 0.250 s, I used the equation [tex]x=x_0+v_0t+\frac{1}{2}at^2[/tex], where x_0 and v_0 are both 0. so x = 1/2g(.250s)^2 = 0.307m.

I then said that at 0.250s, [tex]x_{COM}=\frac{(2m)(0)+(m)(0.307)}{m+2m}=0.012meters[/tex]

So at 0.600s, the position of the COM is given by [tex]x=x_0+v_0t+\frac{1}{2}at^2[/tex]. I solved that saying using 0.012m as the initial position and 0m/s as the initial velociy, and got an incorrect answer (1.87m)

PART B:
Not surprisingly, when I used my answer for part (a) in the equation [tex]v^2=v_0^2+2a\Delta x[/tex], I also got the wrong anwer.

And now I am stuck (and feeling stupid, because this problem got one out of three dots for difficulty).
 
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  • #2
For part A, you should simply find where the 2 stones are at .6 sec, and find their center of mass.
 
  • #3
Thanks, that got the right answer for part a.
For part b, I wrote the equation for the position COM purely symbolically then took the first derivative and used that to find velocity. So that's that!
 

1. What is the center of mass?

The center of mass is a point in a system of objects where the entire mass of the system can be considered to be concentrated. It is the point at which the system will balance, and it is also the average position of all the individual masses in the system.

2. How is the center of mass calculated?

The center of mass is calculated by taking the sum of the product of each individual mass and its distance from a chosen reference point, then dividing by the total mass of the system. This can be expressed mathematically as:

(xcm = ∑mixi / ∑mi)
(ycm = ∑miyi / ∑mi)
(zcm = ∑mizi / ∑mi)

3. How does the center of mass affect the motion of objects?

The center of mass plays a crucial role in determining the motion of objects. In a system of objects, the center of mass will move in a straight line at a constant velocity unless acted upon by an external force. This means that the motion of the individual objects in the system will be influenced by the motion of the center of mass.

4. What happens to the center of mass when objects are in motion?

The center of mass will continue to move in a straight line at a constant velocity as long as no external forces act upon the system. However, if external forces are present, the center of mass may accelerate or change direction. Additionally, the relative positions of the individual objects within the system may affect the position of the center of mass.

5. How does the center of mass relate to stability?

The location and motion of the center of mass are important factors in determining the stability of an object or system. Objects with a low center of mass are generally more stable than those with a high center of mass. Additionally, a system's center of mass must remain within its base of support in order for the system to remain stable. If the center of mass falls outside of the base of support, the system will become unstable and may topple over.

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