/*! 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 122 A truck is hauling a 300-kg log ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 122

A truck is hauling a 300-kg log out of a ditch using a winch attached to the back of the truck. Knowing the winch applies a constant force of 2500 N and the coefficient of kinetic friction between the ground and the log is 0.45, determine the time for the log to reach a speed of 0.5 m/s.

Short Answer

Expert verified
The log reaches 0.5 m/s in approximately 0.13 seconds.

Step by step solution

01

Calculate the Force of Friction

First, determine the force of kinetic friction which opposes the motion of the log. The force of friction \( f_k \) is given by \( f_k = \mu_k \times N \), where \( \mu_k = 0.45 \) is the coefficient of kinetic friction and \( N = m \times g \) is the normal force. Assume \( g = 9.81 \text{ m/s}^2 \). So, \( N = 300 \times 9.81 = 2943 \text{ N} \). Therefore, \( f_k = 0.45 \times 2943 = 1324.35 \text{ N} \).
02

Determine the Net Force

The net force \( F_{net} \) required to accelerate the log is the force applied by the winch minus the frictional force: \( F_{net} = 2500 - 1324.35 = 1175.65 \text{ N} \).
03

Calculate the Acceleration

Use Newton's second law \( F = m \times a \) to find the acceleration \( a \) of the log. Rearranging gives \( a = \frac{F_{net}}{m} = \frac{1175.65}{300} = 3.92 \text{ m/s}^2 \).
04

Determine the Time Required to Reach the Desired Speed

Use the kinematic equation \( v = u + at \), where \( v = 0.5 \text{ m/s} \) is the final speed, \( u = 0 \text{ m/s} \) is the initial speed, and \( t \) is the time. Solving for \( t \) gives \( t = \frac{v - u}{a} = \frac{0.5 - 0}{3.92} \approx 0.13 \text{ s} \).

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Key Concepts

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

Newton's Second Law
Newton's second law provides a fundamental framework for understanding how forces influence the motion of objects. The law is typically stated as \( F = ma \), where \( F \) represents the net force acting on an object, \( m \) is the mass of the object, and \( a \) is the acceleration. In the context of our exercise, where a 300-kg log is being moved, this law allows us to calculate the acceleration needed for the log to achieve a desired speed given the forces acting on it.

When the winch applies a force of 2500 N, this force is opposed by the friction between the ground and the log. The net force is the difference between these two forces. Once we determine the net force, we use Newton's second law to calculate the acceleration of the log. The accurate calculation of this acceleration is crucial for further determining how quickly the log reaches a specific speed.
Kinematic Equations
Kinematic equations are essential tools in physics used to describe the motion of objects. These equations relate four key variables: velocity, acceleration, displacement, and time. In our exercise, the kinematic equation \( v = u + at \) is used to ascertain the amount of time it takes for the log to accelerate from rest to a speed of 0.5 m/s.

Here, \( v \) is the final velocity (0.5 m/s), \( u \) is the initial velocity (0 m/s, since the log starts from rest), \( a \) is the acceleration we calculated using Newton's second law (3.92 m/s虏), and \( t \) is the unknown time we solve for. With this understanding, the equation allows us to rearrange and solve for \( t \), providing the time needed for the log to reach the desired speed. This application fully showcases how kinematic equations are used to predict motion over time given a set of motion parameters.
Coefficient of Friction
The coefficient of friction is a crucial value in physics that measures how much friction force exists between two surfaces. In our problem, the coefficient of kinetic friction, \( \mu_k = 0.45 \), describes the interaction between the log and the ground as the log moves.

This coefficient is used in the formula \( f_k = \mu_k \times N \) to calculate the kinetic friction force \( f_k \), where \( N \) is the normal force. For the log, the normal force is equal to the weight of the log, which is the mass (300 kg) multiplied by the gravitational force (9.81 m/s虏).
  • Determining the force of friction is critical because it opposes the winch's force and impacts the net force available for acceleration.
  • The greater the coefficient of friction, the more resistance there is to motion.
Understanding the coefficient of friction provides insight into how different surfaces can affect movement and is pivotal for solving problems involving frictional forces.

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Most popular questions from this chapter

A 1.4-kg model rocket is launched vertically from rest with a constant thrust of 25 N until the rocket reaches an altitude of 15 m and the thrust ends. Neglecting air resistance, determine (a) the speed of the rocket when the thrust ends, (b) the maximum height reached by the rocket, (c) the speed of the rocket when it returns to the ground.

A boy located at point \(A\) halfway between the center \(\mathrm{O}\) of asemicircular wall and the wall itself throws a ball at the wall in adirection forming an angle of \(45^{\circ}\) with \(O A .\) Knowing that after hitting the wall the ball rebounds in a direction parallel to \(O A,\) determine the coefficient of restitution between the ball and the wall.

The \(1-\mathrm{kg}\) ball at \(A\) is suspended by an inextensible cord and given an initial horizontal velocity of \(5 \mathrm{m} / \mathrm{s}\). If \(l=0.6 \mathrm{m}\) and \(x_{\mathrm{B}}=0,\) determine \(y_{\mathrm{B}}\) so that the ball will enter the basket.

A 500-g collar can slide without friction along the semicircular rod BCD. The spring is of constant 320 N/m and its undeformed length is 200 mm. Knowing that the collar is released from rest at B, determine (a) the speed of the collar as it passes through C, (b) the force exerted by the rod on the collar at C.

A ballistic pendulum is used to measure the speed of high-speed projectiles. A \(6-\mathrm{g}\) bullet \(A\) is fired into a \(1-\mathrm{kg}\) wood block \(B\) suspended by a cord with a length of \(l=2.2 \mathrm{m}\). The block then swings through a maximum angle of \(\theta=60^{\circ} .\) Determine \((a)\) the initial speed of the bullet \(v_{0},(b)\) the impulse imparted by the bullet on the block, \((c)\) the force on the cord immediately after the impact.
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