Cube & Pivoting Rod Collision

In summary: Don't forget that v1 is negative.In summary, the cube of mass m slides without friction at a speed vo and undergoes a perfectly elastic collision with a rod of length d and mass 2m that is pivoted about a frictionless axle through its center. Using conservation of energy and conservation of momentum, the velocity of the cube after the collision is found to be (1/5)vo in the opposite direction of its initial velocity.
  • #1
bcjochim07
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0

Homework Statement


A cube of mass m slides without friction at a speed vo. It undergoes a perfectly elastic collision with the bottom tip of a rod length d and mass 2m. The rod is pivoted about a frictionless axle through its center, and initially it hangs straight down and is at rest. What is the cube's velocity- both speed and direction after the collision?


Homework Equations


Moment of inertia for rod pivoted about center I=(1/12)mr^2


The Attempt at a Solution



I used conservation of energy

Kcubeinitial = Krotational of rod + Kcubefinal
I replaced angular velocity with v1/.5d

.5mvo^2 = .5mvf^2 + .5(1/12)2md^2 * (v1/.5)^2

Then I used conservation of momentum:
mvo= mvf + 2mv1

But substituting these two equations into each other leads to something I can't solve. Should I try angular momentum?
 
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  • #2
Hi bcjochim07,

The problem with using conservation of linear momentum is that the rod does not have a single linear speed--the tip is going fastest, near the pivot is going slow, halfway along the rod it's going half the speed of the tip, etc.

But the entire rod has the same angular speed, so conservation of angular momentum is useful here.
 
  • #3
Ok, so for angular momentum

mvo*(d/2) = mvf(d/2) + I*omega

v= omega * r omega= v/(d/2)

But I haven't been get the expression for the velocity of the cube yet, which is just a multiple of vo
 
  • #4
You'll have to use both conservation of kinetic energy and conservation of angular momentum.

Your conservation of kinetic energy formula was

.5mvo^2 = .5mvf^2 + .5(1/12)2md^2 * (v1/.5)^2

but it looks like it's missing a factor of d; omega here is v1/(0.5 d), so the d's will cancel.

Putting this together with your conservation of angular momentum gives two equations in two unknowns (because the answer should be in terms of d and m).
 
  • #5
Angular momentum:
m(d/2)vo = m(d/2)vf + (1/12)(2m)d^2*(v1/.5d)
(d/2)vo= (d/2)vf + (1/3)dv1
(1/2)vo = (1/2)vf + (1/3)v1

Kinetic energy
.5mvo^2 = .5mvf^2 + .5(1/12)(2m)(d^2)(v1/.5d)^2
.5vo^2=.5vf^2 + (1/12) (v1/.5)^2
.5vo^2=.5vf^2 + (1/3) v1^2

but how do I substitute these into each other to come up with answer, (1/5)vo ?
 
  • #6
Hi bcjochim07,

You have two equations in two unknowns vf and v1. You can solve one of the equations (the momentum would probably be easiest) for v1 for example and then plug it into the other equation.

(You'll probably have a quadratic equation to solve unless there is a fortuitous cancellation.)
 

1. What is a cube and pivoting rod collision?

A cube and pivoting rod collision is a type of collision that occurs when a cube-shaped object collides with a pivoting rod-shaped object. This can happen in a physics simulation or in real life when two physical objects come into contact with each other.

2. What factors determine the outcome of a cube and pivoting rod collision?

The outcome of a cube and pivoting rod collision is determined by several factors, including the mass, velocity, and angle of approach of the objects, as well as the material and shape of the objects.

3. How does the shape of the objects affect the collision?

The shape of the objects can greatly affect the collision. For example, a cube colliding with a pivoting rod at a perpendicular angle will result in different outcomes than if the cube collides at an angle. The shape also affects the distribution of force and energy during the collision.

4. Can a cube and pivoting rod collision be perfectly elastic?

Yes, a cube and pivoting rod collision can be perfectly elastic if the objects are made of materials that can store and release energy without any loss. However, in most real-life scenarios, some energy is usually lost due to factors such as friction and deformation of the objects.

5. How is a cube and pivoting rod collision relevant in real-world applications?

Cube and pivoting rod collisions have real-world applications in areas such as engineering, sports, and transportation. Understanding how these collisions occur and how they can be manipulated can help in the design and development of various products, structures, and machines.

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