Analysis of Two Connected Uniform Rods: Dynamics & Forces

[CaptFormal]

Homework Statement

Two uniform rods are connected to a table by pivots at one end. Rod B is longer than rod A. Both are released simultaneously from an initial angle θ. Neglect air friction. NOTATION: CM = center of mass; α = angular acceleration; |ay|= size of downward acceleration.

1. αA and αB are dependent on θ.
2. The density of the rods affect their rate of fall.
3. |ax| of the CM initially equals 0 for both rods.
4. αA and αB both increase with time.
5. Rods A and B hit the table at the same time.
6. αA and αB are the same initially.
7. Just before landing, the CM of B has a greater speed than the CM of A.
8. |ay| is initially equal for the CM of A and B.

Homework Equations

This is a multiple choice type of answer which one must choose from the following:

-True
-False
-Decreases
-Increases
-Stays the Same

The Attempt at a Solution

Well the first thing that threw me off was the "Decreases", "Increases", and "Stays the Same" answer because they do not even fit in the sentence.

I was thinking that number (1) would be true because angular acceleration depends upon angular velocity which depends on theta (or an angle).

Number (2), assuming that because there is no air resistance then the rods will have to be in a vacuum. Thus, density will have no effect.

Number (3), not sure about this one.

Number (4), angular acceleration does depend on time by the following formula: (change in angular velocity) / (change in time). Thus, the more time that is taken, the less the angular acceleration is. Thus, the answer would be false.

And I am not sure about (5), (6), (7), or (8).

Any help will be appreciated. Thanks.

-Captformal

 

[anathema]

Dear Captformal,

Thank you for your response to the forum post. Here are my thoughts on the statements:

1. True - As you mentioned, angular acceleration is dependent on the initial angle θ, so it would vary for both rods.

2. False - As you correctly pointed out, in a vacuum, the density of the rods would not affect their rate of fall.

3. Stays the Same - This statement is referring to the horizontal acceleration of the center of mass (CM) of both rods. Since there is no horizontal force acting on the rods, the acceleration in the horizontal direction would remain constant at 0.

4. False - As you stated, angular acceleration would decrease over time as the rods fall towards the table.

5. False - Since rod B is longer, it would have a greater distance to travel and would therefore hit the table later than rod A.

6. True - Since both rods are released from the same initial angle, their initial angular acceleration would be the same.

7. True - As the rods fall towards the table, the longer rod (B) would have a greater speed at the CM compared to the shorter rod (A).

8. True - Both rods are released from the same initial angle and have the same downward acceleration, so their CMs would have the same downward acceleration as well.

I hope this helps clarify some of the statements for you. Let me know if you have any further questions.

 

[tomcue]

Your thought process for number (1) is correct, as angular acceleration is dependent on the initial angle θ for both rods.

For number (2), the density of the rods does not affect their rate of fall in a vacuum, so the answer would be "Stays the Same".

Number (3) is also true, as the center of mass of both rods will initially have no horizontal acceleration.

For number (4), you are correct in thinking that the answer is false, as angular acceleration will decrease as time increases.

Number (5) is also true, as both rods are released simultaneously and have the same acceleration due to gravity, so they will hit the table at the same time.

For number (6), the statement is not entirely clear, but if it means that the initial angular acceleration for both rods is the same, then the answer would be "True".

For number (7), this statement is also not entirely clear, but assuming it means that the center of mass of rod B has a greater speed than rod A just before landing, then the answer would be "True".

And for number (8), the statement is true, as both rods have the same downward acceleration due to gravity, so the center of mass of both rods will have the same downward acceleration.