A 1.00 kg hollow ball with a radius of 2.2^10^-1 m, filled with air, is released from rest at the bottom of a 1.50 m deep pool of water. How high above the water does the ball shoot upward if the upward force exerted by the fluid on the ball 40 N? Neglect all frictional effects, and neglect the ball's motion when it is only partially submerged.
I used energy consideration & got the following answer. But the answer is incorrect.Please point out where i have gone wrong
Under the water you have the net force on the ball; F= 40 N. That force acts over
a distance D (1.5 m in this case) and thus does work on the ball giving it some
KE as it leaves the water: KE = FD. = 40*1.5 = 60 J
To find out how high the ball rises in the air, set that initial KE to the PE at the top
of the motion: KE = FD = 60 J = mgH. Now solve for H. where m = 1 kg
H = 60/1*9.8
= 6.12 m
Perhaps it has something to do with the radius of the ball. When the center of the ball has risen 1.50 - 0.44 m, it is no longer fully submerged. They want you to assume that work stops then. It is not clear where to measure the potential energy change from: the surface or one radius below.
The answer you get depends upon the approximations made
To calculate the height to which the ball rises, you're correct in using the principle of conservation of energy. However, it seems like there is a mistake in your calculations.
The work done by the upward force exerted by the fluid on the ball is given by the product of the force and the distance over which it acts. In this case, the force is 40 N, and the distance is 1.5 m, so the work done is:
Work = Force * Distance = 40 N * 1.5 m = 60 J (as you correctly calculated)
Since the ball is initially at rest at the bottom of the pool, it has no kinetic energy, so the total initial energy is equal to the work done by the upward force:
Initial Energy = Work Done = 60 J
As the ball rises above the water, it gains potential energy (PE) equal to its weight multiplied by the height it rises (H). The weight of the ball is given by its mass (m) multiplied by the acceleration due to gravity (g):
Weight = m * g = 1 kg * 9.8 m/s^2 = 9.8 N
Setting the initial energy equal to the potential energy at the top of the motion, we have:
Initial Energy = PE at top = Weight * Height
60 J = 9.8 N * H
Solving for H:
H = 60 J / 9.8 N ≈ 6.12 m
So, your calculation for the height is correct. It seems like there might be a typo or another mistake in the given answer you provided. The correct answer should be approximately 6.12 meters.