/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 148 The truck has a mass of \(50 \ma... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 148

The truck has a mass of \(50 \mathrm{Mg}\) when empty. When it is unloading \(5 \mathrm{~m}^{3}\) of sand at a constant rate of \(0.8 \mathrm{~m}^{3} / \mathrm{s}\), the sand flows out the back at a speed of \(7 \mathrm{~m} / \mathrm{s}\), measured relative to the truck, in the direction shown. If the truck is free to roll, determine its initial acceleration just as the load begins to empty. Neglect the mass of the wheels and any frictional resistance to motion. The density of sand is \(\rho_{s}=1520 \mathrm{~kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
The step-by-step solution shows the method to calculate the initial acceleration of an unloading truck using principles of conservation of momentum and Newton's second law of motion. After calculation the mass flow rate of the sand, the momentum before and after unloading starts of the system, the final step is to equate the momentum change rate of the sand to the force acting on the truck to find its initial acceleration.

Step by step solution

01

- Calculate the mass flow rate of sand

First calculate the mass flow rate of the sand using the given data. The mass of the sand flowing per unit time, also known as mass flow rate, is given by the product of the density of the sand, the volume flow rate or the rate at which the sand is flowing. Using the formula \(mass flow rate= density 脳 flow rate\), where density = \(1520 kg/m^3\) and flow rate = \(0.8 m^3/s\)
02

- Calculate the momentum of the system before unloading started

The next step is to calculate the total momentum of the system before the unloading process begins. The momentum before the sand starts to unload is the product of the total mass of the truck (including the sand) and its initial velocity. The mass of the truck and the sand together is given, which is \(50 Mg = 50000 Kg\) and the initial velocity is 0 (as it鈥檚 stationary initially) which makes the total initial momentum = 50000 Kg * 0 = 0
03

- Calculate the momentum of the system as unloading begins

Next, calculate the momentum of the system when the sand starts to unload. The sand is exiting the truck at a rate of \(0.8 m^3/s\) and a speed of \(7 m/s\) and the truck is still stationary. The momentum of the system is now the vector sum of the momentum of the sand and the truck. \n\nThe momentum of the unloaded sand is the product of mass flow rate and velocity of sand which we have calculated earlier.\n\nAfter that, to calculate the momentum of the truck we will subtract the momentum of the sand from the initial momentum of the truck i.e (0 - momentum of sand) as these two momenta direct opposite to each other. Hence the initial acceleration of the truck is \(a= -momentum_{sand} / mass_{truck} \)
04

- Calculate the initial acceleration

Since the problem places no restrictions on movement of the truck, the body of the truck itself moves in response to the sand being unloaded. Apply Newton's second law \(F=ma\) here to find the initial acceleration of the truck. As the force acting on the truck can be taken as the momentum change rate of the sand (force = rate of change of momentum), the acceleration of truck is then the force (which is the momentum change rate of the sand) divided by the mass of the truck.

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Most popular questions from this chapter

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