Projectile hits rod hanging from pivot.

In summary, a thin, uniform bar weighing 90N is hanging vertically from the ceiling. It is hit by a 3kg ball traveling at 10 m/s, which rebounds in the opposite direction at 6 m/s. Using the equations for angular momentum and moment of inertia, the change in the angular momentum of the ball is equal to the change in the angular momentum of the rod. The correct formula for the moment of inertia of the rod is I = 1/12 * MR^2.
  • #1
BOYLANATOR
198
18

Homework Statement


A thin, uniform bar, 2, long and weighing 90N is hanging vertically from the ceiling by a frictionless pivot. It is struck by a small 3kg ball, 1.5m below the ceiling, initially traveling horizontally at 10 m/s. The ball rebounds in the opposite direction with a speed of 6 m/s.

Homework Equations



Lbefore = Lafter

L = Iω

Irod = [itex]\frac{1}{2}[/itex]MR2

Ipoint = MR2

The Attempt at a Solution



At the point of impact the ball can be thought of as a particle in circular motion about the pivot with radius 1.5m and tangential velocity 10m/s.
The change in velocity is 16m/s. So the effective change in the angular momentum of the ball is

ΔLball = Iball Δωball = [itex]\frac{IballΔv}{Rball}[/itex]

This is equal to the change in the angular momentum of the rod (opposite direction):

ΔLrod = Irod Δωrod = [itex]\frac{IballΔv}{Rball}[/itex]

Insert values for I and rearrange :

Δωrod = (2*mball*rball*Δvball) / (mrod*rball2)

This gives ω = 3.92 rad/s , the given answer is 5.88 rad/s.
 
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  • #2
I obviously used the fraction syntax incorrectly, please let me know what I did wrong. Thanks
 
  • #3
BOYLANATOR said:

Homework Statement


A thin, uniform bar, 2, long and weighing 90N is hanging vertically from the ceiling by a frictionless pivot. It is struck by a small 3kg ball, 1.5m below the ceiling, initially traveling horizontally at 10 m/s. The ball rebounds in the opposite direction with a speed of 6 m/s.


Homework Equations



Lbefore = Lafter

L = Iω

Irod = [itex]\frac{1}{2}[/itex]MR2

The formula for the moment of inertia of the rod is not correct.


ehild
 
  • #4
BOYLANATOR said:
I obviously used the fraction syntax incorrectly, please let me know what I did wrong
itex gives up if you put non-itex codes like [s u b] inside. Use ^ for sup and _ for sub.
 
  • #5
As haruspex said, _ for sub, but enclose subscript between curly thingies {} :smile:

[itex]ΔL_{ball} = I_{ball} Δω_{ball} = \frac{I_{ball}Δv}{R_{ball}}[/itex]

written as

ΔL_{ball} = I_{ball} Δω_{ball} = \frac{I_{ball}Δv}{R_{ball}}

ehild
 
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  • #6
Ah yes. I should stick to deriving the moments of inertia, my memory doesn't serve me well. Thanks.
 

1. How does the angle of the projectile affect where it hits the rod?

The angle of the projectile will determine the horizontal distance from the pivot point where it hits the rod. A higher angle will result in a longer horizontal distance, while a lower angle will result in a shorter horizontal distance.

2. Does the mass of the projectile affect the impact on the rod?

Yes, the mass of the projectile will affect the force of the impact on the rod. A heavier projectile will have a greater force and may cause the rod to move more.

3. What factors besides angle and mass can affect the projectile's impact on the rod?

The speed of the projectile, air resistance, and the length and material of the rod can also affect the impact. The speed and air resistance will determine the trajectory of the projectile, while the length and material of the rod will affect the force of the impact.

4. Can the pivot point be moved to change where the projectile hits the rod?

Yes, moving the pivot point will change the location where the projectile hits the rod. This is because the pivot point determines the center of rotation for the rod, and the projectile will follow a curved path around this point.

5. Is there a specific formula for calculating the impact of a projectile on a hanging rod?

Yes, the formula for calculating the impact of a projectile on a hanging rod is: F = (m * v^2 * sin(2θ))/L, where F is the force of the impact, m is the mass of the projectile, v is the velocity of the projectile, θ is the angle of the projectile, and L is the length of the rod.

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