Answer:
M[min] = M[basket+people+ balloon, not gas] * ΔR/R[b]
ΔR is the difference in density between the gas inside and surrounding the balloon.
R[b] is the density of gas inside the baloon.
====================================
Let V be the volume of helium required.
Upthrust on helium = Weight of the volume of air displaced = Density of air * g * Volume of helium = 1.225 * g * V
U = 1.225gV newtons
----
Weight of Helium = Volume of Helium * Density of Helium * g
W[h] = 0.18gV N
Net Upward force produced by helium, F = Upthrust - Weight = (1.225-0.18) gV = 1.045gV N -----
Weight of 260kg = 2549.7 N
Then to lift the whole thing, F > 2549.7
So minimal F would be 2549.7
----
1.045gV = 2549.7
V = 248.8 m^3
Mass of helium required = V * Density of Helium = 248.8 * 0.18 = 44.8kg (3sf)
=====
Let the density of the surroundings be R
Then U-W = (1-0.9)RgV = 0.1RgV
So 0.1RgV = 2549.7 N
V = 2549.7 / 0.1Rg
Assuming that R is again 1.255, V = 2071.7 m^3
Then mass of hot air required = 230.2 * 0.9R = 2340 kg
Notice from this that M = 2549.7/0.9Rg * 0.1R so
M[min] = Weight of basket * (difference in density between balloon's gas and surroundings / density of gas in balloon)
M[min] = M[basket] * ΔR/R[b]
Pour the entire components into water.
First the iron filings can be separated using a magnet as iron is a magnetic element.
By pouring the mixture into water, it allows the salt to dissolve in the water, while the sand will not.
Next we can run this solution through a filtration device to separate the sand from the water.
Now all that's left is to let the water evaporate so that the salt will be exposed that dissolved into the water.
Answer:
The angular velocity at the beginning of the interval is
.
Explanation:
Given that,
Angular acceleration 
Angular displacement 
Angular velocity 
We need to calculate the angular velocity at the beginning
Using formula of angular velocity


Where,
= angular acceleration
= angular velocity
Put the value into the formula



Hence, The angular velocity at the beginning of the interval is
.
There are some missing data in the text of the problem. I've found them online:
a) coefficient of friction dry steel piston - steel cilinder: 0.3
b) coefficient of friction with oil in between the surfaces: 0.03
Solution:
a) The force F applied by the person (300 N) must be at least equal to the frictional force, given by:

where

is the coefficient of friction, while N is the normal force. So we have:

since we know that F=300 N and

, we can find N, the magnitude of the normal force:

b) The problem is identical to that of the first part; however, this time the coefficienct of friction is

due to the presence of the oil. Therefore, we have:
As stated in the statement, we will apply energy conservation to solve this problem.
From this concept we know that the kinetic energy gained is equivalent to the potential energy lost and vice versa. Mathematically said equilibrium can be expressed as


Where,
m = mass
= initial and final velocity
g = Gravity
h = height
As the mass is tHe same and the final height is zero we have that the expression is now:





