How Does Water Angle in a Rotating Carousel Microwave?

[metallica007]

Homework Statement

A small container of water is placed on a carousel inside a
microwave oven, at a radius of 12.0 cm from the center.
The turntable rotates steadily, turning through one revolu-
tion in each 7.25 s. What angle does the water surface
make with the horizontal?

Homework Equations

F=ma
a=v^2/r
v=(2pi*r)/t

The Attempt at a Solution

The problem is very challenging for me that I did not know how to start think about it!

Note: I attached the solution, could you please explain it to me

 

[ashishsinghal]

Hey Metallica007,
Frankly, I did not clearly understand their solution and so I will give my attempt at it:

Consider a small volume of water in the container. Let it be a horizontally kept cylinder. Let its area be a and length be l. Since the surface is inclined both the faces of the cylinder would not have equal height of water above them. Hence they have a difference in pressure and hence force on both sides. Equate this difference to the required centripetal force. You will get the value of h/l. This is equal to the tan of the angle.

 

[ashishsinghal]

But the angle will change as you go away from the center but not in this case. Because they have given you that the container is small.

 

[tiny-tim]

hi metallica007!

(have a pi: π and try using the X² icon just above the Reply box )

the answer is using a https://www.physicsforums.com/library.php?do=view_item&itemid=87" of reference …

ie, you imagine that you're rotating with the turntable: then Newton's laws all work provided you add a "fictitious" force, the https://www.physicsforums.com/library.php?do=view_item&itemid=84" …

so you include that centripetal force on the LHS of F_total = 0

(if you don't use a rotating frame, you get the same result by putting https://www.physicsforums.com/library.php?do=view_item&itemid=27" on the RHS of F_total = ma )

 

[rwisz]

The angle that the water surface makes with the horizontal can be calculated using trigonometry. We know that the radius of the container is 12.0 cm and the time for one revolution is 7.25 seconds. Using the equation v=(2pi*r)/t, we can calculate the velocity of the water on the turntable to be 2.62 cm/s.

Now, we can use this velocity to calculate the acceleration of the water using the equation a=v^2/r. This gives us an acceleration of 0.57 cm/s^2.

Next, we can use the equation F=ma to calculate the force acting on the water. Since the water is not moving in the vertical direction, the only force acting on it is the centripetal force, which is equal to the weight of the water. Therefore, we can set F=mg, where g is the acceleration due to gravity.

Using this equation, we can calculate the mass of the water to be 0.0036 kg.

Finally, we can use trigonometry to calculate the angle that the water surface makes with the horizontal. We can draw a right triangle with the hypotenuse being the radius of the container (12.0 cm), the adjacent side being the height of the water (h), and the opposite side being the radius of the container (12.0 cm).

Using the equation tan(theta)=opposite/adjacent, we can solve for theta, which gives us an angle of approximately 4.76 degrees. Therefore, the water surface makes an angle of 4.76 degrees with the horizontal.