Find the magnitude of the initial acceleration of the rod's center of mass

In summary, the problem involved finding the magnitude of the initial acceleration of a uniform rod attached to a frictionless pivot at one end, released from rest at an angle of 17 degrees above the horizontal. Using the equation torque=I*angular acceleration and the moment of inertia for a rod, the angular acceleration was found to be 12.22 rad/s^2. Then, using the formula a_t=r*alpha, the tangential acceleration of the center of mass was calculated to be 14.05 m/s^2. However, it was later discovered that the angle used in the calculation was incorrect and needed to be adjusted by adding 90 degrees. After redoing the calculations, the correct answer was obtained.
  • #1
fruitl00p
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Homework Statement



A uniform rod of length 1.15m is attached to a frictionless pivot at one end. It is released from rest an angle theta=17.0 degrees above the horizontal.

Find the magnitude of the initial acceleration of the rod's center of mass

Homework Equations



torque=moment of inertia*angular acceleration
torque=length*force(perpendicular)

I=1/3 ML^2

The Attempt at a Solution



I set the torque about pivot equal to the moment of inertia about pivot times the angular acceleration to find the angular acceleration. Then I found the linear acceleration by multiplying the found angular acceleration by the length of rod. So my work was...

L/2*(Mg)cos17=1/3*ML^2*(angular acceleration)
3/2*(g/L)*cos17=(angular acceleration)
12.22 rad/s^2= (angular acceleration)

a=(angular acceleration)*L= 14.05 m/s^2

However the answer I got is wrong. (Unfortunately, I do not have access to the correct answer) What step did I miss?
 
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  • #2
... L/2*(Mg)cos17=1/3*ML^2*(angular acceleration)

This should be the sin not the cos.

The com will then experience only a tangential acceleration:

[tex]a_t = r \alpha[/tex]
 
  • #3
I replaced the cos with the sin17 in my equation and I just tried to solve but I was told my answer was wrong. And I did exactly what you suggested. I wonder why that was wrong? Did I need to do something else?
 
  • #4
Have you used the correct angle ? 17 deg above the horizontal is 90 + 17 from the vertical.
 
  • #5
If you are trying to find the tangential acceleration of the centre of mass, wouldn't it be [tex]a_t = \alpha(L/2)[/tex], since the centre of mass will be at half the length of a uniform rod? Just a thought.
 
  • #6
Mentz114, thank you for mentioning that! I looked through some notes and saw I need to add 90 degrees. However, I just redid the problem, using the info you told me and what andrevdh told me, and oddly enough, I got the same answer!

hage567, I will try to find the initial acceleration with L/2 instead of L.
 
  • #7
Thanks everyone, I got the right answer now.
 

1. What does "magnitude of the initial acceleration" mean?

The magnitude of the initial acceleration refers to the size or strength of the acceleration experienced by the rod's center of mass at the beginning of its motion.

2. How is the initial acceleration of the rod's center of mass measured?

The initial acceleration of the rod's center of mass can be measured by tracking the displacement of the center of mass over a short period of time and using the formula a = (vf - vi)/t, where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.

3. What factors affect the magnitude of the initial acceleration of the rod's center of mass?

The magnitude of the initial acceleration of the rod's center of mass is affected by the force applied to the rod, the mass of the rod, and the distribution of mass along the rod.

4. Can the initial acceleration of the rod's center of mass be negative?

Yes, the initial acceleration of the rod's center of mass can be negative if the rod is decelerating or changing direction.

5. Why is finding the magnitude of the initial acceleration important in scientific research?

Finding the magnitude of the initial acceleration of the rod's center of mass is important because it helps to understand the dynamics of the system and can be used to make predictions about the future motion of the rod. It is also a fundamental concept in physics and is used in many real-world applications, such as designing machines and structures that can withstand certain forces and accelerations.

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