/*! 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 14 A dump truck is being filled wit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 14

A dump truck is being filled with sand. The sand falls straight downward from rest from a height of \(2.00 \mathrm{~m}\) above the truck bed, and the mass of sand that hits the truck per second is \(55.0 \mathrm{~kg} / \mathrm{s}\). The truck is parked on the platform of a weight scale. By how much does the scale reading exceed the weight of the truck and sand?

Short Answer

Expert verified
The scale reading exceeds by 344.3 N due to the momentum change of the falling sand.

Step by step solution

01

Calculate the velocity of the sand when it hits the truck

The sand falls from rest under the influence of gravity from a height of 2.00 m. Using the equation for the final velocity of a freely falling object, we have:\[ v = \sqrt{2gh} \]where \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity and \( h = 2.00 \text{ m} \) is the height. Substituting these values, we get:\[ v = \sqrt{2 \times 9.81 \times 2} = \sqrt{39.24} = 6.26 \text{ m/s} \]
02

Determine the momentum change per second

The momentum change per second is the impulse, which equals the force exerted by the sand on the truck. Since mass per second \( \dot{m} \) is \( 55.0 \text{ kg/s} \) and velocity \( v = 6.26 \text{ m/s} \), the impulse is \( F = \dot{m} \times v \):\[ F = 55.0 \times 6.26 = 344.3 \text{ N} \]
03

Calculate the excess force on the scale

The scale measures this force exerted by the momentum change as an additional weight. The force \( 344.3 \text{ N} \) due to the sand changing velocity equals the additional force the scale needs to support beyond the static weight of the truck and sand.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Momentum
Momentum is a fundamental concept in physics, representing the quantity of motion an object has. It's essential to understand momentum when analyzing collisions and impacts.
Momentum is calculated as the product of an object's mass and velocity. Mathematically, it's expressed as:
  • Momentum (\( p \)) = mass (\( m \)) 脳 velocity (\( v \))
A key point about momentum is that it's a vector quantity, which means it has both a magnitude and direction. This is crucial in scenarios involving changes in the direction of motion.
When sand falls onto the truck, it gains velocity due to gravity, and thus gains momentum. Since the sand impacts the truck continually, understanding how momentum changes over time helps us calculate forces involved, like how much extra force the truck scale reads.
Impulse
Impulse is closely related to momentum. It describes the change in momentum over time and is directly related to the force applied over a specific period.
Mathematically, impulse (\( J \)) is given by:
  • Impulse = Change in Momentum (\( riangle p \))
  • Impulse = Force (\( F \)) 脳 Time (\( riangle t \))
In the case of the dump truck, impulse explains the interaction between the falling sand and the truck. Each second, 55 kg of sand hits the truck, changing its velocity as it impacts. This results in a force, calculated as impulse, that the truck's scale reads as additional weight.
Understanding impulse helps us acknowledge how swiftly an object's motion can change and how these changes exert forces in dynamic systems.
Free Fall
Free fall occurs when an object falls solely under the influence of gravity, with no other forces acting on it, like friction or air resistance.
The velocity of a freely falling object can be determined using the formula:
  • \( v = \sqrt{2gh} \)
where \( v \) is the final velocity, \( g \) is the acceleration due to gravity, and \( h \) is the height from which it falls.
In our problem, the sand starts from rest and falls 2 meters, acquiring velocity by the time it hits the truck. This falling motion is critical as it determines the speed and momentum of the sand hitting the truck, contributing to the forces calculated.
Recognizing the principles of free fall allows us to solve real-world problems like this one, where understanding the speed and impact of falling objects is necessary for determining forces.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A mine car (mass \(=440 \mathrm{~kg}\) ) rolls at a speed of \(0.50 \mathrm{~m} / \mathrm{s}\) on a horizontal track, as the drawing shows. A \(150-\mathrm{kg}\) chunk of coal has a speed of \(0.80 \mathrm{~m} / \mathrm{s}\) when it leaves the chute. Determine the velocity of the car/coal system after the coal has come to rest in the car.

At illustrates the physics principles in this problem. An astronaut in his space suit and with a propulsion unit (empty of its gas propellant) strapped to his back has a mass of \(146 \mathrm{~kg}\). During a space-walk, the unit, which has been completely filled with propellant gas, ejects some gas with a velocity of \(+32 \mathrm{~m} / \mathrm{s}\). As a result, the astronaut recoils with a velocity of \(-0.39 \mathrm{~m} / \mathrm{s}\). After the gas is ejected, the mass of the astronaut (now wearing a partially empty propulsion unit) is \(165 \mathrm{~kg}\). What percentage of the gas propellant in the completely filled propulsion unit was depleted?

ssm A golf ball bounces down a flight of steel stairs, striking several steps on the way down, but never hitting the edge of a step. The ball starts at the top step with a vertical velocity component of zero. If all the collisions with the stairs are elastic, and if the vertical height of the staircase is \(3.00 \mathrm{~m}\), determine the bounce height when the ball reaches the bottom of the stairs. Neglect air resistance.

Consult Interactive Solution \(\underline{7} .9\) at for a review of problem- solving skills that are involved in this problem. A stream of water strikes a stationary turbine blade horizontally, as the drawing illustrates. The incident water stream has a velocity of \(+16.0 \mathrm{~m} / \mathrm{s},\) while the exiting water stream has a velocity of \(-16.0 \mathrm{~m} / \mathrm{s}\). The mass of water per second that strikes the blade is \(30.0 \mathrm{~kg} / \mathrm{s}\). Find the magnitude of the average force exerted on the water by the blade.

One average force \(\overline{\vec{F}}_{1}\) has a magnitude that is three times as large as that of another average force \(\overline{\mathrm{F}}_{2} .\) Both forces produce the same impulse. The average force \(\overline{\mathrm{F}}_{1}\) acts for a time interval of \(3.2 \mathrm{~ms}\). For what time interval does the average force \(\overline{\mathrm{F}}_{2}\) act?
See all solutions

Recommended explanations on Physics Textbooks

Dynamics

Read Explanation

Measurements

Read Explanation

Particle Model of Matter

Read Explanation

Wave Optics

Read Explanation

Classical Mechanics

Read Explanation

Magnetism

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.