What Determines When a Metal Ball Leaves an Oscillating Tray?

[iwonde]

Homework Statement

A 1.50-kg, horizontal, uniform tray is attached to a vertical ideal spring of force constant 185 N/m and a 275-g metal ball is in the tray. The spring is below the tray, so it can oscillate up-and-down. The tray is then pushed down 15.0 cm below its equilibrium point (call this point A) and released from rest. (a) How high above point A will the tray be when the metal ball leaves the tray? (b) How much time elapses between releasing the system at point A and the ball leaving the tray? (c) How fast is the ball moving just as it leaves the tray?

Homework Equations

ΣF = ma_y
-mg-ky = m((d^2y)/(dt^2)) + (k/m)y + g = 0

The Attempt at a Solution

I'm not sure of how to approach this problem, but I'm thinking of solving for y in the equation above.

 

[DavidWhitbeck]

I'll provide a hint:

When the ball reaches the position at which it leaves the tray, it must lose contact with the tray, that implies that the condition for leaving the tray is that the normal force vanishes. You can solve for the normal force on the ball using Newton's 2nd Law.

The first thing you should then do is draw a free body diagram, identify all forces acting on the ball and in which direction they act.

After that you can write down the form of Newton's 2nd Law that holds for the ball.

After that apply my hint to solve for the position at which the ball leaves the tray, and then do the usual song and dance with your periodic motion equations, and I'm sure by that point you'll have no trouble proceeding.

 

[Moetasim]

Since the tray is initially pushed down 15.0 cm, the initial displacement (y_0) would be -0.15 m. Using this information, I can then solve for the amplitude of the oscillation (A) by setting the potential energy at point A equal to the kinetic energy at the highest point (point B) when the ball leaves the tray. This will give me the amplitude of the oscillation, which is the maximum height the tray will reach.

Once I have the amplitude, I can use the equation for simple harmonic motion (SHM) to solve for the time it takes for the tray to reach point B. From there, I can calculate the speed of the ball just as it leaves the tray by using the equation for velocity in SHM.

However, I also need to take into consideration the mass of the ball and how it affects the motion of the tray. Since the ball is attached to the tray, it will also contribute to the oscillation. I can solve for the total mass of the system (m_T) by adding the mass of the tray (m_t) and the mass of the ball (m_b). This will give me a new equation for SHM, taking into account the total mass of the system.

By solving for the amplitude, time, and velocity of the ball using the new equation, I can get a more accurate solution for the problem. Additionally, I can also use the equation for the period of SHM to find the time it takes for the ball to leave the tray, which will also give me the time elapsed between releasing the system at point A and the ball leaving the tray.

Overall, I would approach this problem by first solving for the amplitude of the oscillation using energy conservation, then using SHM equations to solve for the time and velocity of the ball. I would also take into account the total mass of the system to get a more accurate solution.