How can the end of a rod accelerate faster than g?

In summary, when the right support is removed from a uniform rod, the atom on the right end will accelerate faster than ##g## at ##1.5g## due to the interatomic forces of attraction acting horizontally. However, this does not necessarily mean that the net force on the atom is able to make it accelerate faster than ##g##. This is because the forces between the atoms are not assumed to be horizontal, as a rod can support shear stress as well as tension. Therefore, the end of the rod falls down faster than ##g## because of the additional downward shear forces acting on it.
  • #1
Happiness
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A uniform rod rests on supports at its ends. The right support is quickly removed.
Screen Shot 2016-12-13 at 4.57.45 AM.png

The atom on the right end will accelerate faster than ##g## at ##1.5g##. How is this possible forces-wise? The only forces acting on the atom is its own weight and the interatomic forces of attraction, which acts horizontally (left) at the instant the right support is removed. So the net force on the atom doesn't seem to be able to make it accelerate faster than g.
 
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  • #2
Happiness said:
The only forces acting on the atom is its own weight and the interatomic forces of attraction, which acts horizontally (left)
Why do you think those interatomic forces act horizontally?
 
  • #3
Try this: Imagine a small piece of the rod that is just above the right support. What forces act on it?
 
  • #4
Doc Al said:
Why do you think those interatomic forces act horizontally?

Because the rod is horizontal at first.

Suppose we picture the rod to be made of N horizontal rows of atoms. We may group each vertical column of N atoms as a "super-atom". Then these super-atoms are horizontally next one another. So the forces between them are assumed to be horizontal.
 
  • #5
Happiness said:
Because the rod is horizontal at first.

Suppose we picture the rod to be made of N horizontal rows of atoms. We may group each vertical column of N atoms as a "super-atom". Then these super-atoms are horizontally next one another. So the forces between them are assumed to be horizontal.
Just because they are arranged horizontally does not mean that the forces are horizontal. (See the question in my last post.)
 
  • #6
Doc Al said:
Try this: Imagine a small piece of the rod that is just above the right support. What forces act on it?

Suppose the rod is two-atom thick. Let ##M## and ##m## be the masses of the rod and of each atom respectively. Before the right support is removed, there is an upward force of ##\frac{1}{2}Mg## acting on the rightmost column of atoms. The total weight of this column is ##2mg##. Thus to keep it in equilibrium, the adjacent column must be acting on it a downward force of ##F=\frac{1}{2}Mg-2mg##. When the right support is removed, the net force acting on it will be ##F+2mg=\frac{1}{2}Mg##. Its acceleration will thus be ##\frac{Mg}{4m}##. But this is not necessarily ##1.5g##. What's wrong?
 
  • #7
The point of that exercise was not to solve for the acceleration, but just to point out that one section of the rod must be exerting vertical forces on the other.

To solve for the acceleration of the end of the rod we'd just examine the torque on the rod. (The downward force on that end piece would change once the support is removed.)
 
  • #8
Happiness said:
So the forces between them are assumed to be horizontal.
This is not correct.

You are thinking of a rope. An ideal rope can only support tension, meaning a force along the rope. However a rod can support shear stress as well, meaning forces perpendicular to the rod.

The end of the rod falls down faster than g because there is a downward shear force as well as the downward gravitational force.
 
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FAQ: How can the end of a rod accelerate faster than g?

1. How can the end of a rod accelerate faster than g?

This phenomenon is known as angular acceleration and it occurs when the rod is rotating around a fixed point. The end of the rod is traveling a larger distance in the same amount of time as the rest of the rod, resulting in a faster acceleration.

2. What factors affect the end of a rod accelerating faster than g?

The length of the rod, the mass distribution along the rod, and the speed of rotation are all factors that can affect the end of a rod accelerating faster than g.

3. Why does the end of a rod accelerate faster than g in a centrifuge?

In a centrifuge, the rod is rotating around a central axis at a high speed. This creates a centrifugal force that pulls the end of the rod towards the outer edge, resulting in a faster acceleration than g.

4. Is it possible for the end of a rod to accelerate slower than g?

Yes, it is possible for the end of a rod to accelerate slower than g. This can happen if the rod is rotating at a slower speed or if there is a counteracting force, such as friction, acting against the rotation.

5. How is angular acceleration different from linear acceleration?

Angular acceleration is the rate of change of angular velocity, which is the speed at which an object is rotating around a fixed point. Linear acceleration, on the other hand, is the rate of change of linear velocity, which is the speed at which an object is moving in a straight line. These two types of acceleration are different because they involve different types of motion - rotation versus linear movement.

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