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Chapter 4: Problem 87

Sand is released from the hopper \(H\) with negligible velocity and then falls a distance \(h\) to the conveyor belt. The mass flow rate from the hopper is \(m^{\prime} .\) Develop an expression for the steady-state belt speed \(v\) for the case \(h=0 .\) Assume that the sand quickly acquires the belt velocity with no rebound, and neglect friction at the pulleys \(A\) and \(B\).

Short Answer

Expert verified
The belt must move with speed \(v = 0\) for steady state when \(h = 0\).

Step by step solution

01

Understand the problem

We are asked to develop an expression for the belt speed \(v\) in a system where sand is released from a hopper directly onto a conveyor belt. The sand has negligible initial velocity, quickly acquires the belt velocity, and the distance \(h\) from the hopper to the belt is zero. Also, friction at the pulleys is neglected.
02

Apply Conservation of Momentum

Since the sand has negligible initial velocity and quickly acquires the speed of the belt, we apply the conservation of linear momentum to the system. The momentum change experienced by the sand equals the momentum imparted by the belt. Thus, \(m' \cdot 0 + m' \cdot v = m' \cdot v\), where \(m'\) is the mass flow rate and \(v\) is the speed of the belt.
03

Derive the Expression for Belt Speed

Because the distance \(h\) is zero, there are no complications from falling distance. The belt imparts velocity to the sand instantaneously as it lands on the belt. By the conservation of momentum, if initially the sand had no velocity, then the belt speed must equal the sand speed after they interact. Thus, for \(h = 0\), the belt must move with the speed \(v = 0\) to keep the system in steady state, which implies no net momentum change.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Flow Rate
Mass flow rate denotes the amount of mass passing through a given point in a system per unit of time. It is often represented by the symbol \(m^{\prime}\) and is considered constant for steady systems. In the context of the sand falling onto the conveyor belt, \(m^{\prime}\) quantifies the rate at which sand is released from the hopper onto the belt.
Understanding mass flow rate is crucial as it helps determine how much mass the belt must continuously accommodate, influencing the conveyor's operational speed.
  • If the mass flow rate increases, the system must adapt to accommodate the additional mass, potentially requiring a change in belt speed.
  • A consistent mass flow rate ensures a steady operation, maintaining the desired process flow without interruptions.
Steady-State
In physics and engineering, steady-state refers to a condition where system properties like mass, energy, and velocity remain constant over time. For the problem at hand, we aim for a steady-state where the conveyor belt and the sand it carries move at a constant speed.
This means:
  • There is no net change in the mass or speed of sand on the belt over time.
  • The mass flow rate \(m^{\prime}\) is constant, so the belt speed must remain constant to maintain this state.
Achieving steady-state is crucial as it simplifies system analysis and ensures predictable, efficient system behavior.
Negligible Velocity
Negligible velocity means that the initial speed of the sand, as it leaves the hopper, is so small that it can be effectively considered zero. In this exercise, this assumption simplifies calculations and focuses on the belt and sand interaction's essential dynamics.
  • The assumption that the velocity is negligible means initial sand momentum is practically zero.
  • This simplifies the application of the conservation of momentum, as all the velocity energy is derived from the belt.
By starting with negligible velocity, the problem highlights how the belt's speed alone accelerates the sand to match its motion without initial momentum influence.
Belt Speed
The belt speed, denoted by \(v\), determines how quickly the belt moves and consequently how quickly the sand begins to move at the same speed. In the context of this exercise:
  • The belt speed is influenced directly by the mass flow rate \(m^{\prime}\) and the requirement to achieve steady-state conditions.
  • For our specific problem, \(h = 0\) means the sand lands directly on the belt without falling, making it crucial for the belt to immediately impart its speed to the sand.
Using the conservation of momentum, the belt needs to move with a specific speed to ensure no net change in system momentum; thus, its speed equalizes the speed needed by the sand as it merges with the belt motion.

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Most popular questions from this chapter

A horizontal bar of mass \(m_{1}\) and small diameter is suspended by two wires of length \(l\) from a carriage of mass \(m_{2}\) which is free to roll along the horizontal rails. If the bar and carriage are released from rest with the wires making an angle \(\theta\) with the vertical, determine the velocity \(v_{\text {ble }}\) of the bar relative to the carriage and the velocity \(v_{c}\) of the carriage at the instant when \(\theta=0 .\) Neglect all friction and treat the carriage and the bar as particles in the vertical plane of motion.

The upper end of the open-link chain of length \(L\) and mass \(\rho\) per unit length is released from rest with the lower end just touching the platform of the scale. Determine the expression for the force \(F\) read on the scale as a function of the distance \(x\) through which the upper end has fallen. (Comment: The chain acquires a free-fall velocity of \(\sqrt{2 g x}\) because the links on the scale exert no force on those above, which are still falling freely. Work the problem in two ways: first, by evaluating the time rate of change of momentum for the entire chain and second, by considering the force \(F\) to be composed of the weight of the links at rest on the scale plus the force necessary to divert an equivalent stream of fluid.)

At a bulk loading station, gravel leaves the hopper at the rate of 220 lb/sec with a velocity of \(10 \mathrm{ft} / \mathrm{sec}\) in the direction shown and is deposited on the moving flatbed truck. The tractive force between the driving wheels and the road is \(380 \mathrm{lb}\), which overcomes the 200 lb of frictional road resistance. Determine the acceleration \(a\) of the truck 4 seconds after the hopper is opened over the truck bed, at which instant the truck has a forward speed of \(1.5 \mathrm{mi} / \mathrm{hr}\) The empty weight of the truck is \(12,000 \mathrm{lb}\).

A small rocket-propelled vehicle weighs \(125 \mathrm{lb}\), in cluding 20 lb of fuel. Fuel is burned at the constant rate of 2 lb/sec with an exhaust velocity relative to the nozzle of \(400 \mathrm{ft} / \mathrm{sec}\). Upon ignition the vehicle is released from rest on the \(10^{\circ}\) incline. Calculate the maximum velocity \(v\) reached by the vehicle. Neglect all friction.

A jet of fresh water under pressure issues from the \(3 / 4\) -in.-diameter fixed nozzle with a velocity \(v=120 \mathrm{ft} / \mathrm{sec}\) and is diverted into the two equal streams. Neglect any energy loss in the streams and compute the force \(F\) required to hold the vane in place.
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