Classical Mechanics Accelerated Frame/Rotation Problem

In summary, Sally is conducting an experiment on a playground ride where she throws a tennis ball in a way that it passes through the axis of rotation and returns to her. The ball's path is limited to a horizontal plane and air resistance and the Earth's rotation are neglected. Sally must aim the ball at a specific angle in order for it to pass through the axis. The problem involves finding the ball's position and velocity in both the Earth's frame and the rotating frame.
  • #1
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Homework Statement



Sally the physics student conducts the following experiment: There is a popular playground ride which is just a horizontal wooden disk free to rotate around a vertical axis. Sally hops onto the disk (spinning counter clock wise with angular velocity [tex]\omega[/tex] ) with a bunch of tennis balls. With practice, Sally discovers that she can throw a tennis ball such that the path of the ball passes through the axis and returns to her.

Throughout this problem neglect air resistance and neglect the vertical component of the ball's velocity (ie pretend the ball's path is limited to a horizontal plane). You may also neglect any effects due to the rotation of the earth.

A. Sketch a path of the ball r (t) in Earth frame. Choose the coordinate system to coincide with axis of ride.

(I think this is just a straight chord)

B. Find an expression for r(t) in Earth frame. Give an answer in terms of Sally's distance from the axis R and speed of the ball v. (Hint check that v(t) is constant)

(I got just r(t) = R + vt) Too simple?

C. Find the speed of the ball v in terms of R and [tex]\omega[/tex] assuming that the ball passes through the axis and returns to Sally.

(HELP??)

D. Transform the coordinates to find x'(t) and y'(t), the path of the ball in the rotating frame centered on the axis and co-spinning with the ride. Sketch the path of the ball in the rotating frame.

(need the prior answer)

E. Take the time derivative of r'(t) to find the components of the ball's velocity in the rotating frame. Check that v(t)=v'(t) + [tex]\omega[/tex] x r' (t).

(need prior answer)

Homework Equations



a = a' + (omegadot) x r' + 2(omega) x v' + (omega) x (omega x r')
v = v' + (omega) x r' + v0
 
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  • #2
Find the initial velocity components of the tennis ball with respect to the Earth's frame keeping in mind an initial tangential velocity component exists due to rotation.
 
  • #3
thanks! I'll try working it again.
 
  • #4
Okay so I got r_(t) = R + vt + Rwt
which is good, because v_(t) = v+Rw

Now how can I find the value for v in part C?
 
  • #5
Express the ball position in terms of unit vector components

[tex]\vec{r}=A\hat{x}+B\hat{y}[/tex]

where A and B are the magnitudes, and let r(t) = 0 for the ball passing through the axis of rotation with respect to the Earth's reference frame. This is one equation with two unknowns. Also, express Sally's position in terms of vector components. Then equate Sally's position to the position of the ball. This will give two equations with two unknowns. Think about how Sally has to release the ball. Does she aim directly at the center of rotation in order for the ball to pass through this center?
 
  • #6
Thanks for the help - I'll get back to that problem
 
  • #7
cool - i got v = 2rw, which makes enough sense to me.
 

1. What is an accelerated frame in classical mechanics?

An accelerated frame in classical mechanics refers to a frame of reference that is undergoing a change in velocity or direction. This change in velocity can be caused by a force acting on the frame, such as gravity or friction.

2. How does an accelerated frame affect the laws of motion?

An accelerated frame can affect the laws of motion, particularly Newton's laws, by introducing apparent forces known as fictitious forces. These forces only appear to act on an object within the accelerated frame, and are not present in an inertial frame of reference.

3. What is the Coriolis effect and how does it relate to rotation in classical mechanics?

The Coriolis effect is a phenomenon that occurs in rotating frames of reference, including the Earth. It causes objects moving in a straight line to appear to curve due to the rotation of the frame. This effect is important in classical mechanics, as it must be taken into account when analyzing the motion of objects in rotating frames.

4. How do you solve a rotation problem in classical mechanics?

To solve a rotation problem in classical mechanics, you must first identify the forces acting on the rotating object and their respective torques. You can then use Newton's second law of motion for rotational motion, along with the equations for torque and angular momentum, to determine the angular acceleration and final angular velocity of the object.

5. Why is understanding classical mechanics important for scientists?

Understanding classical mechanics is crucial for scientists as it forms the basis for many other areas of physics, such as electromagnetism and quantum mechanics. It also helps us to understand and predict the behavior of objects in everyday life, from the motion of planets to the movement of machines and vehicles.

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