How to Calculate the Work Done When Pushing a Ball Underwater?

[Drektjal]

Homework Statement

A ball of which radius is 15 inches is put into water and it sinks 2.5 inches. How much is the work done when it's pushed halfway under water?

Homework Equations

Air pressure doesn't have to be taken into account (not sure about that but that one I can do myself anyways).

Lift by the water (the weight of the ball can be calculated with this):

$$N=\rho Vg$$

Average V of the part of the ball that is underwater is (this one I made up myself):

$$V_{Av}=\frac{ \sum^{\infty}_{n=0} V_{n}}{\infty}$$

Where$$V_{n}$$ is the volume of the ball underwater when the ball is up to its $$\frac{n}{\infty}*r$$ underwater (where r is the radius of the ball).

Work done:

$$W= \rho V_{Av}g * 15 inches - mg * 15 inches$$ (Where m is the mass of the ball and W is the final answer. )

The Attempt at a Solution

I can do everything else in this exercise except I can't calculate $$V_{Av}=\frac{\sum ^{\infty}_{n=0} V_{n}}{\infty}$$ (the hard part is getting the $$\infty$$ out of the expression) but I'm not even sure if it's supposed to be calculated this way as we haven't been taught the concept of $$\sum$$.

This exercise is one of the preliminary exercises that came with my physics book so it's not homework or something like that and therefore I'm not interested in the exact answer but rather the theory and easiest way to solve it.

Thanks in advance for any kind of help.

 

[HallsofIvy]

Homework Statement

A ball of which radius is 15 inches is put into water and it sinks 2.5 inches. How much is the work done when it's pushed halfway under water?

Homework Equations

Air pressure doesn't have to be taken into account (not sure about that but that one I can do myself anyways).

Since it says you don't have to worry about that I imagine you can!

Lift by the water (the weight of the ball can be calculated with this):

$$N=\rho Vg$$

Average V of the part of the ball that is underwater is (this one I made up myself):

$$V_{Av}=\frac{ \sum^{\infty}_{n=0} V_{n}}{\infty}$$

Where$$V_{n}$$ is the volume of the ball underwater when the ball is up to its $$\frac{n}{\infty}*r$$ underwater (where r is the radius of the ball).

The Attempt at a Solution

I can do everything else in this exercise except I can't calculate $$V_{Av}=\frac{\sum ^{\infty}_{n=0} V_{n}}{\infty}$$ (the hard part is getting the $$\infty$$ out of the expression) but I'm not even sure if it's supposed to be calculated this way as we haven't been taught the concept of $$\sum$$.

This exercise is one of the preliminary exercises that came with my physics book so it's not homework or something like that and therefore I'm not interested in the exact answer but rather the theory and easiest way to solve it.

Thanks in advance for any kind of help.

You can't do it as a sum and you don't have "infinity" in it- if you want to do it exactly you will need to use an integral, not a sum.

 

[Drektjal]

You can't do it as a sum and you don't have "infinity" in it- if you want to do it exactly you will need to use an integral, not a sum.

-So you'd recon there's no other way around this than that as integral is also something that hasn't been taught to us (if I remember right I don't have the integral-class in maths until next semester)?

Thanks for the quick answer though!