Calculating Maximum Volume of Sand in a Circular Area

[winlinux]

1)A hot-air balloon of mass M is descending vertically with downward acceleration of magnitude a. How much mass (ballast) must be thrown out to give the balloon an upward acceleration of magnitude a? Assume that the upward force from the air (the lift) does not change because of the decrease in mass.

3)fig3
A worker wishes to pile a cone of sand onto a circular area in his yard. The radius of the circle is R, and no sand is to spill onto the surrounding area. If s is the static coefficient of friction between each layer of sand beneath it, show that the greatest volume of sand that can be stored in this manner is . (fig3)

 

[Martin Cook]

Newton's second law

Hi

The figures didn't seem to work for your second question, but here's my attempt at an answer for the first.

First, choose downwards to be positive. Call the upwards force due to the hot air F_{up}, and the original mass of the balloon m_i. Therefore

m_1 g - F_{up} = m_1 a
m_1 = \frac{F_{up}}{g-a}.

(g is gravitational acceleration, therefore m_1 g is the weight.)

Call the new mass after the ballast has thrown out m_2. The new acceleration is -a.

<br /> m_2 g - F_{up} = - m_2 a
<br /> m_2 = \frac{F_{up}}{g+a}<br />.

The difference in the mass is m_1 - m_2, which is

<br /> m_1-m_2 = F_{up}\left( \frac{1}{g-a}-\frac{1}{g+a}\right)

<br /> m_1-m_2 = F_{up} \frac{2a}{g^2-a^2}<br />

 

[wais]

To calculate the maximum volume of sand that can be stored in a circular area with a radius of R, we first need to determine the height of the cone of sand that can be piled without spilling onto the surrounding area.

Using the given information, we know that the worker must overcome the static friction between each layer of sand in order to pile the cone. This means that the maximum height of the cone can be achieved when the weight of the sand equals the maximum frictional force that the worker can exert.

We can express this relationship as:

Ff = μs * mg

where Ff is the maximum frictional force, μs is the static coefficient of friction, m is the mass of the sand, and g is the acceleration due to gravity.

Since the worker wishes to pile the sand without it spilling, the weight of the sand must be equal to the maximum frictional force, which we can express as:

mg = μs * mg

Simplifying, we get:

m = μs * m

This means that the mass of the sand must be equal to the product of the static coefficient of friction and the mass of the sand.

Now, we can use the formula for the volume of a cone to calculate the maximum volume of sand that can be stored in the circular area:

V = 1/3 * π * R^2 * h

where V is the volume, π is the constant pi, R is the radius, and h is the height of the cone.

Plugging in the value for m from earlier, we get:

V = 1/3 * π * R^2 * (μs * m)

Simplifying, we get:

V = μs * (1/3 * π * R^2 * m)

Since we know that the mass of the sand must be equal to μs * m, we can substitute this into the equation and get:

V = μs * (1/3 * π * R^2 * μs * m)

Simplifying further, we get:

V = 1/3 * μs^2 * π * R^2 * m

Therefore, the greatest volume of sand that can be stored in a circular area with a radius of R is given by:

V = 1/3 * μs^2 * π * R^2 * m

This formula can be used to calculate the maximum volume of sand that can be stored