/*! 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 40 You find it takes 200 \(\mathrm{... [FREE SOLUTION] | 91Ó°ÊÓ

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You find it takes 200 \(\mathrm{N}\) of horizontal force to move an empty pickup truck along a level road at a speed of 2.4 \(\mathrm{m} / \mathrm{s}\) . You then load the pickup and pump up its tires so that its total weight increases by 42\(\%\) while the coefficient of rolling friction decreases by 19\(\%\) . Now what horizontal force will you need to move the pickup along the same road at the same speed? The speed is low enough that you can ignore air resistance.

Short Answer

Expert verified
229.32 N is needed to move the loaded pickup truck.

Step by step solution

01

Identify the original conditions

Initially, the horizontal force required to move the empty pickup truck along the road at a speed of 2.4 m/s is 200 N.
02

Understand the new conditions

The total weight of the pickup truck increases by 42%, and the coefficient of rolling friction decreases by 19% after loading the truck and inflating its tires.
03

Establish the relationship between force, weight, and rolling friction

The force required to move the truck depends on the rolling friction, which is calculated as the product of the weight of the truck and the coefficient of rolling friction:\[F = \text{weight} \times \text{coefficient of rolling friction}\]
04

Calculate the new weight

Let the initial weight of the truck be \(W\). After adding 42%, the new weight \(W'\) is:\[W' = W + 0.42W = 1.42W\]
05

Calculate the new coefficient of rolling friction

Let the initial coefficient of rolling friction be \(\mu\). After decreasing by 19%, the new coefficient of rolling friction \(\mu'\) is:\[\mu' = \mu - 0.19\mu = 0.81\mu\]
06

Calculate the initial force of friction

The initial force of friction is given:\[F = W \times \mu = 200 \text{ N}\]
07

Solve for new force to move the truck using the new conditions

The new force of friction \(F'\) can be calculated using the adjusted weight and coefficient of friction:\[F' = W' \times \mu' = (1.42W) \times (0.81\mu)\]\[F' = 1.42 \times 0.81 \times (W \times \mu)\]\[F' = 1.42 \times 0.81 \times 200 \text{ N}\]
08

Perform the calculation

Calculate the new force:\[F' = 1.42 \times 0.81 \times 200 = 229.32 \text{ N}\]

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Key Concepts

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

Rolling Friction
Rolling friction is a type of resistance that objects encounter while they roll on a surface. Unlike sliding friction, which occurs when surfaces slide against each other, rolling friction is generally much lower.

When a wheel, ball, or similar object rolls over a surface, there isn't direct surface-to-surface dragging. Instead, deformation of the wheel or the surface, or sometimes both, occurs.

This deformation results in energy loss, contributing to rolling friction. Factors affecting rolling friction include:
  • The nature of surfaces in contact
  • The material of the rolling object
  • The extent to which the surfaces deform
  • The smoothness of surfaces
Understanding rolling friction is important when analyzing mechanical systems like vehicles or machinery, where minimizing energy loss due to friction can enhance performance and efficiency.
Coefficient of Friction
The coefficient of friction is a dimensionless number that represents how much frictional force exists between two surfaces. It is crucial for calculating friction forces and for predicting how easily one surface will move over another.

There are two main types of coefficients:
  • Static Friction Coefficient: for objects not in motion
  • Kinetic Friction Coefficient: for moving objects
For rolling objects, like a car tire, we use the rolling friction coefficient, which is generally less than the static or kinetic friction coefficients.

In our problem, reducing the coefficient of rolling friction by 19% reflects changes in conditions, such as inflating tires, which hardens the surface and reduces deformation. This calculation involves determining how the coefficient changes the rolling friction force, and finally how the rolling friction responds to weight changes.
Physics Problem Solving
Physics problem solving involves applying fundamental principles to analyze and solve situational problems systematically. In our given exercise, solving for the new required horizontal force involves:

1. **Understanding the Parameters**: Knowing the initial and changed conditions – such as the weight and the coefficient of rolling friction.
2. **Applying the Formulas**: Using \[F = ext{weight} \times ext{coefficient of rolling friction}\]to calculate relevant forces.
3. **Sequential Calculations**: - Calculating the new weight \(W'\) by increasing the initial weight by 42%. - Determining the new coefficient of rolling friction \(\mu'\) by decreasing the initial by 19%. - Finally, finding the new force using the adjusted values.
4. **Performing the Computation**: Ensuring calculations are done correctly to predict the outcome, leading to solutions like finding the new force of 229.32 N.
Problem-solving leads to strengthening your understanding of physics and improving your analytical abilities, which are necessary for tackling real-world challenges.
Horizontal Force
Horizontal force refers to any force acting parallel to the surface of the ground. In dynamics and statics problems, understanding and accurately determining the horizontal force is central to analyzing motion.

In our truck problem, horizontal force is the actual force exerted to overcome rolling friction and keep the truck moving at a constant speed. Given the force value of 200 N initially, changes to the truck's weight and rolling friction coefficient lead to recalculating the required effort for fluency in movement.
  • Changes in weight and friction coefficients alter the amount of horizontal force needed.
  • Less rolling frictions means less force is necessary because less opposition occurs against movement.
By calculating this horizontal force, you understand its effect on the motion of vehicles and can make adjustments necessary for designing mechanisms or systems with efficient movement strategies.

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