Find Initial Acceleration of Rod's Center of Mass with Newton's 2nd Law

In summary, the question is asking for the magnitude of the initial acceleration of a uniform rod that is released from rest at an angle of 27 degrees above the horizontal. The solution involves using Newton's Second Law of rotation, Τ = I α, to determine the rotational acceleration. Since the rod is constrained to rotate about the hinge, the only force creating a torque is the object's weight. By solving for α algebraically, the linear acceleration can be found as rα.
  • #1
jstealth03
1
0
here's the question:

A uniform rod of length 1.15 m is attached to a frictionless pivot at one end. It is released from rest from an angle of 27 degrees above the horizontal. Find the magnitude of the initial acceleration of the rod's center of mass.

So far I've only been able to figure out that I have to use Newton's Second Law of rotation, F x r = I x alpha. How do i derive that to figure out acceleration? Do i break down F into ma, and cancel the m's on each side that result from inertia?
 

Attachments

  • prob28_1012rod.gif
    prob28_1012rod.gif
    1.5 KB · Views: 303
Physics news on Phys.org
  • #2
Since the rod is constrained to rotate about the hinge, all you need is Newton's 2nd law for rotation. That's: Τ = I α.

The only force creating a torque about the pivot is the object's weight. You'll also need to know the rotational inertia of the rod.

So, solve for α. The linear acceleration is rα.

Do things algebraically and you'll find that things will cancel nicely.
 
  • #3


To find the initial acceleration of the rod's center of mass, we can use Newton's Second Law of rotation, which states that the net torque acting on an object is equal to its moment of inertia times its angular acceleration. In this case, the net torque acting on the rod is due to the force of gravity, which is equal to the weight of the rod acting at its center of mass. Therefore, we can write the equation as:

τ = Iα

Where τ is the net torque, I is the moment of inertia, and α is the angular acceleration.

To find the moment of inertia for a uniform rod rotating about one end, we can use the formula I = (1/3)ML^2, where M is the mass of the rod and L is its length. Plugging in the values given in the question, we get I = (1/3)(M)(1.15)^2.

Now, we can substitute this value into our equation and solve for α:

τ = Iα
Mg(1.15/2)sin(27) = (1/3)(M)(1.15)^2α
(0.575M)sin(27) = (0.0391M)α
α = (0.575sin(27))/(0.0391) ≈ 1.23 rad/s^2

This is the angular acceleration of the rod. To find the linear acceleration of the center of mass, we can use the formula a = rα, where r is the distance from the pivot to the center of mass. In this case, r = 1.15/2 = 0.575 m.

Thus, the initial acceleration of the rod's center of mass is:

a = (0.575)(1.23) ≈ 0.707 m/s^2

In conclusion, the magnitude of the initial acceleration of the rod's center of mass is approximately 0.707 m/s^2.
 

1. What is Newton's 2nd Law and how does it relate to finding the initial acceleration of a rod's center of mass?

Newton's 2nd Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this case, we can use this law to find the initial acceleration of a rod's center of mass by analyzing the forces acting on the rod and its mass.

2. What is the center of mass and why is it important in this scenario?

The center of mass is the point where the entire mass of an object is concentrated. In this scenario, it is important because it allows us to simplify the analysis of the rod's motion and determine the initial acceleration of the rod as a whole.

3. What are the steps to finding the initial acceleration of a rod's center of mass using Newton's 2nd Law?

The steps include analyzing the forces acting on the rod, determining the net force, calculating the mass of the rod, and using the formula a = F/m to find the initial acceleration of the rod's center of mass.

4. What are some common forces that may act on a rod and how do they affect the initial acceleration of the center of mass?

Some common forces that may act on a rod include gravity, friction, tension, and applied forces. These forces can either increase or decrease the initial acceleration of the center of mass, depending on their magnitude and direction.

5. Can the initial acceleration of a rod's center of mass change over time?

Yes, the initial acceleration of a rod's center of mass can change over time if the forces acting on the rod change. For example, if an additional force is applied to the rod or if friction increases, the initial acceleration of the rod's center of mass will also change.

Similar threads

  • Introductory Physics Homework Help
2
Replies
42
Views
3K
  • Introductory Physics Homework Help
Replies
7
Views
637
  • Introductory Physics Homework Help
Replies
5
Views
316
  • Introductory Physics Homework Help
Replies
13
Views
858
  • Introductory Physics Homework Help
2
Replies
38
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
478
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
821
Back
Top