How Do You Calculate the Radial Force on a Hinged Stick?

• justagirl
In summary, mechanics is the branch of physics that studies motion and forces in physical objects. The main principles of mechanics are described by Newton's laws of motion, and there is a distinction between statics (objects at rest or in equilibrium) and dynamics (objects in motion). A force is a push or pull that can affect an object's motion or shape, and mechanics is applied in everyday activities like driving, biking, and using machines and structures.
justagirl

Can someone please help me with this problem? I've been working on it for hours - anything would help thanks!

A system consists of a uniform stick of length L and mass M hinged at one end. The hinge is released from rest at an angle theta_0 with respect to the vertical. Show that the radial force exerted on the stick F = Mg/2*(5 cos theta - 3 cos (theta_0), where theta is the angle of the stick with resect to the vertical after it is released.

I tried to solve the problem using conservation of energy.

I said potential energy is equal to the height that the center of mass has fallen, so I got:

[(L/2)(sin_theta - sin theta_0)]Mg = 1/2Iw^2 = (1/2)(1/3)MR^2*w^2
((L/2)sin_theta - sin theta_0)g = (1/6)R^2w^2
[3L (sin_theta - sin theta_0)g] / R^2 = w^2

v = wr, v^2 = w^2 * r^2

v^2 = [3L (sin_theta - sin theta_0)g]

F = Mv^2 / R, which is not what they had.

HELP!

Here's a couple of points that might help:

First, your coordinate system is not quite right. The problem is posed in terms of the angle with respect to the vertical so all your sines should be cosines.

Second, you're not dealing with a point mass so you will need the moment of inertia for the stick in your energy calculation.

Finally, the radial force on the stick consists of two parts. One is the component of the gravitational force along the stick and the other is the reaction force of the hinge which is the negative of the centripetal force.

Putting it altogether I get:

$$F_r = mg\cos \theta - \frac {mv_{\theta}^{2}}{L/2}$$

You should find

$$\frac {mv_{\theta}^{2}}{L/2} = \frac {3mg}{2} \left(\cos \theta_0 - \cos \theta \right)$$

and the correct result follows.

thanks

Everything did come together. Thank you for your help!

Suppose now that I want to calculate the tangential force.

I set torque = I(alpha) = F(R)

F = I(alpha) / R = (1/3)ML^2*(a/R) / R

F = (1/3)ML^2*((g sin_theta)/(L/2)) / (L/2)

I got that F_tangential = 4/3Mgsin_theta, but they got that the answer
is 1/4 Mg sin_theta.

What did I do wrong? Thanks!

The tangential force should be $mg \sin \theta$. It's not clear what they did to get the 1/4.

What is mechanics?

Mechanics is the branch of physics that deals with the study of motion and forces, and how they affect the behavior of physical objects.

What are the main principles of mechanics?

The main principles of mechanics are Newton's laws of motion, which describe how objects move and interact with each other under the influence of forces.

What is the difference between statics and dynamics in mechanics?

Statics is the branch of mechanics that deals with the study of objects at rest or in equilibrium, while dynamics deals with objects in motion and the forces that cause them to move.

What is a force in mechanics?

A force is a push or pull that can cause an object to change its state of motion or shape. It is described by its magnitude, direction, and point of application.

How is mechanics used in everyday life?

Mechanics is used in many aspects of our daily lives, such as in driving a car, riding a bike, or throwing a ball. It is also crucial in the design and functioning of machines and structures, such as bridges and buildings.

• Introductory Physics Homework Help
Replies
21
Views
2K
• Introductory Physics Homework Help
Replies
11
Views
902
• Introductory Physics Homework Help
Replies
2
Views
5K
• Introductory Physics Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
13
Views
2K
• Introductory Physics Homework Help
Replies
55
Views
3K
• Introductory Physics Homework Help
Replies
3
Views
311
• Introductory Physics Homework Help
Replies
5
Views
917
• Introductory Physics Homework Help
Replies
21
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
284