How Does Sand Falling Impact Balance in Physics?

[johnconnor]

I am a CIE A Level student and the following question is of 1st year undergraduate level.

A bag containing a mass M of sand is suspended from a hook on the arm of a balance at a height h above the balance pan. At a time t=0, the sand starts to pour from a hole at the bottom of the bagm falling onto the pan beneath, and continues at a constant rate r (mass/unit time) until the bag is empty.

i. Find the mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag (a simple balance with equal arms is envisaged).

ii. Show graphically the variation with time of the mass required to maintain balance throughout the experiment, indicating by suitable labelling the quantities involved. Assume ideal conditions, under which air resistance, balance inertian and damping effects may be ignored.

My working:

When we consider the pouring of sand from a height h above the pan, there exists 3 phases: the falling sand yet to reach the pan; the steady stream of sand whose momentum is destroyed upon reaching the pan and the final stage when the steady stream of sand reduces to zero when all of the sand in the bag is depleted.

So for the "mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag", I tried starting from the conservation of energy gh = 0.5 v^2. Since the height is yet to be found, I couldn't use the impulsive force on destroying of momentum too.

I apologise for not having complex mathematical workings to present. please help. Thank you.

 

[gneill]

I am a CIE A Level student and the following question is of 1st year undergraduate level.

A bag containing a mass M of sand is suspended from a hook on the arm of a balance at a height h above the balance pan. At a time t=0, the sand starts to pour from a hole at the bottom of the bagm falling onto the pan beneath, and continues at a constant rate r (mass/unit time) until the bag is empty.

i. Find the mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag (a simple balance with equal arms is envisaged).

ii. Show graphically the variation with time of the mass required to maintain balance throughout the experiment, indicating by suitable labelling the quantities involved. Assume ideal conditions, under which air resistance, balance inertian and damping effects may be ignored.

My working:

When we consider the pouring of sand from a height h above the pan, there exists 3 phases: the falling sand yet to reach the pan; the steady stream of sand whose momentum is destroyed upon reaching the pan and the final stage when the steady stream of sand reduces to zero when all of the sand in the bag is depleted.

So for the "mass required on the other pan to maintain balance under steady conditions when a continuous stream of sand is falling from the bag", I tried starting from the conservation of energy gh = 0.5 v^2. Since the height is yet to be found, I couldn't use the impulsive force on destroying of momentum too.

I apologise for not having complex mathematical workings to present. please help. Thank you.

Welcome to PF johnconnor.

Here's a diagram to consider:

Note that the mass of the sand is distributed in three places at any given time so long as the sand continues to stream from the sac to the pan. When the falling sand stops at the pan it will deliver up its momentum by way of exerting a force Fp on the pan. You should be able to determine what that force will be based upon the mass-delivery rate of the sand and the velocity it gains as it falls.

 

[johnconnor]

Thank you, gneill. Pardon me if my question sounds elementary, but may I know what questions did you ask yourself when you were attempting the question?

Could you please demonstrate my query with the question below?

An aerator nozzle is fitted 0.40m below the surface of the water in a fish tank. It is noted that air bubbles from the nozzle accelerate upwards for the first few millimetres of their motion but then rise to the surface with a constant terminal speed. For bubbles of diameter 3.0mm, this speed is found to be 0.31m/s. The passage of the bubbles through the water is turbulent.

i. Deduce the value of the constant K, in the expression for the retarding force due to turbulence in this system, expressed as KAv²ρ, where ρ is the liquid density, v is the uniform velocity under turbulent conditions and A is the cross-sectional area of the bubble at right angles to the direction of motion.
(The density of water is taken as 1.0x10³ kg m^-3 and the density of air in the bubbles is 1.3 kg m^-3.)

ii. As the bubbles rise, the pressure due to the water above them decreases. Estimate the resulting fractional change in the raduis of a bubble as it rises from the nozzle to the surface.

Attempt:

i. Condition for resultant force = 0.
i.e., upthrust = weight + retardation force
ρVg = 1.8E-7 + KAv²ρ
Taking 1.8E-7 ≈ 0,
K = rg/v²
Substituting values, K = 0.15

In the answer, the constant K is given the expression 4rg/3v² whereas its value is 0.2. May I ask why is there a 4/3 factor? What did I miss out? Since the bubbles are so small, why is the factor of 4/3 - which I suppose relates to the dimension of a sphere - play a role? What other questions am I supposed to ask myself?

ii. I have no idea how to do. pV = nRT didn't help much; perchance I applied the wrong methods involving the formula.

Your help is greatly appreciated. Thank you.

 

[gneill]

Thank you, gneill. Pardon me if my question sounds elementary, but may I know what questions did you ask yourself when you were attempting the question?

I simply asked myself what forces are present (weights), what sources of energy exist (potential energy), what conditions have to hold to satisfy the problem (system remains balanced). Then I drew a Free Body Diagram for left hand pan.

Could you please demonstrate my query with the question below?

Looks like a whole new problem. In general you should start a new thread for a new problem.

An aerator nozzle is fitted 0.40m below the surface of the water in a fish tank. It is noted that air bubbles from the nozzle accelerate upwards for the first few millimetres of their motion but then rise to the surface with a constant terminal speed. For bubbles of diameter 3.0mm, this speed is found to be 0.31m/s. The passage of the bubbles through the water is turbulent.

i. Deduce the value of the constant K, in the expression for the retarding force due to turbulence in this system, expressed as KAv²ρ, where ρ is the liquid density, v is the uniform velocity under turbulent conditions and A is the cross-sectional area of the bubble at right angles to the direction of motion.
(The density of water is taken as 1.0x10³ kg m^-3 and the density of air in the bubbles is 1.3 kg m^-3.)

ii. As the bubbles rise, the pressure due to the water above them decreases. Estimate the resulting fractional change in the raduis of a bubble as it rises from the nozzle to the surface.

Attempt:

i. Condition for resultant force = 0.
i.e., upthrust = weight + retardation force
ρVg = 1.8E-7 + KAv²ρ
Taking 1.8E-7 ≈ 0,
K = rg/v²
Substituting values, K = 0.15

In the answer, the constant K is given the expression 4rg/3v² whereas its value is 0.2. May I ask why is there a 4/3 factor? What did I miss out? Since the bubbles are so small, why is the factor of 4/3 - which I suppose relates to the dimension of a sphere - play a role? What other questions am I supposed to ask myself?

What forces are acting on the bubble? There's buoyancy force, weight, and retardation. What expressions are associated with each? Can you write them out?

When the conditions stated in the problem are in effect (constant speed), how are the forces related to each other? (Free Body Diagram again).

ii. I have no idea how to do. pV = nRT didn't help much; perchance I applied the wrong methods involving the formula.

Assume constant temperature for the rising bubble. The pressure inside the bubble must equal the pressure in the surrounding water. So what determines the pressure in the water surrounding the bubble?

 

[asteriatic]

I can understand your struggle with this problem and I am happy to provide some guidance. Firstly, let's break down the problem into smaller parts and consider the physics involved.

1. The mass required on the other pan to maintain balance under steady conditions is dependent on the mass of the sand falling from the bag, as well as its velocity. This can be explained by the principle of conservation of momentum, which states that the total momentum of a system remains constant in the absence of external forces. Therefore, the mass required on the other pan will be equal to the mass of sand falling from the bag multiplied by its velocity.

2. To determine the velocity of the sand, we can use the principle of conservation of energy. As you have correctly stated, the potential energy of the sand at height h is converted into kinetic energy as it falls to the pan. Therefore, we can use the equation gh = 0.5mv^2 to find the velocity of the sand.

3. The variation of the mass required to maintain balance throughout the experiment can be shown graphically by plotting the mass of sand falling from the bag (which decreases as the bag empties) against time. This will result in a linear decrease in mass over time, as the sand is falling at a constant rate.

4. It is important to note that in ideal conditions, we can ignore air resistance, balance inertia, and damping effects. This means that the sand will fall in a straight line and there will be no external forces acting on the system.

In conclusion, to solve this problem, you will need to use the principles of conservation of momentum and energy. You can also use the equation gh = 0.5mv^2 to find the velocity of the sand. I hope this helps and good luck with your studies!