# Angle of Displacement of Rod Hit by Bullet in Time

• Gena
In summary, a bullet of mass m collides with a rod of mass M and length L that is freely spinning around a fixed point P. The bullet becomes embedded in the rod, causing the system to undergo small oscillations. The expression for the evolution of the angle formed by the rod and the vertical direction can be found by treating the system as a physical pendulum and using conservation of angular momentum. The initial angular velocity of the rod can be calculated using the relevant equations, but the angle of displacement in time cannot be determined without additional information about the system.
Gena

## Homework Statement

A rod of mass M y length L can spin freely around a fixed point P, as shown in the picture. A bullet of mass m is shot perpendicular to the rod, and after impact, it is embedded into it. After impact the system realizes small oscillations, and that immediately after impact, the bullet had a velocity Vo=5m/s.

Determine the expression that represents the evolution of the angle in radians tat the rod forms with respect to the vertical direction (measured counter-clockwise). The time t is measured in seconds once the impact occurs.
Data:
m=0.2kg
L=1.2m
M=1.5kg

## Homework Equations

L=Iw
P=mv
I cm(rod)=(Ml²)/12

## The Attempt at a Solution

I calculated the initial angular velocity of he rod by using the formulas written in relevant equations, but i don't know how to calculate the angle of displacement in time with that information. The fact that the bullet travels along with the rod after hitting it is really confusing as i don't know if the moment of inertia of the system changes.

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Your attempt at a solution is very sketchy. How did you apply the relevant equations? Specifically,
L=Iw: I assume this is the angular momentum relative to the pivot.
P=mv: This is not really useful here because momentum is not conserved since the rod is not free to move.
I cm(rod)=(Ml²)/12: This is the momentum of inertia relative to the CM. You need the expression relative to the pivot. After the bullet is embedded, you need to add its moment of inertia relative to the pivot.

Treat this as a physical pendulum the moment of inertia of which you can calculate. Use conservation of angular momentum to find the angular speed of the rod+bullet system after the collision is completed. The assumption here is that the bullet is embedded in the rod within a very short time interval and before the rod has a change to move appreciably.

## 1. What is the angle of displacement of a rod hit by a bullet in time?

The angle of displacement of a rod hit by a bullet in time refers to the change in direction of the rod caused by the impact of a bullet. It is measured in degrees or radians and can be calculated using the trigonometric functions of the initial and final positions of the rod.

## 2. How is the angle of displacement affected by the speed and mass of the bullet?

The angle of displacement is directly affected by the speed and mass of the bullet. A faster and heavier bullet will have a greater impact force, resulting in a larger angle of displacement of the rod. This is due to the conservation of momentum, where the force of the bullet is transferred to the rod upon impact.

## 3. Does the angle of displacement depend on the material of the rod?

Yes, the angle of displacement can be affected by the material of the rod. Different materials have different densities, strengths, and elasticities, which can influence the amount of force that is absorbed upon impact. This can result in varying angles of displacement for the same bullet speed and mass.

## 4. How can the angle of displacement be measured experimentally?

The angle of displacement can be measured experimentally by using high-speed cameras to capture the motion of the rod upon impact. The footage can then be analyzed to determine the initial and final positions of the rod, and the angle of displacement can be calculated using trigonometric functions.

## 5. What other factors can affect the angle of displacement of a rod hit by a bullet in time?

Apart from the speed, mass, and material of the bullet and rod, other factors that can affect the angle of displacement include the angle of impact, the shape of the bullet and rod, and the presence of any obstacles or barriers that may alter the trajectory of the bullet or the movement of the rod.

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