/*! 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 41 An airliner lands with a speed o... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 41

An airliner lands with a speed of \(50.0 \mathrm{~m} / \mathrm{s}\). Each wheel of the plane has a radius of \(1.25 \mathrm{~m}\) and a moment of inertia of \(110 \mathrm{~kg} \cdot \mathrm{m}^{2}\). At touchdown, the wheels begin to spin under the action of friction. Each wheel supports a weight of \(1.40 \times 10^{4} \mathrm{~N}\), and the wheels attain their angular speed in \(0.480 \mathrm{~s}\) while rolling without slipping. What is the coefficient of kinetic friction between the wheels and the runway? Assume that the speed of the plane is constant.

Short Answer

Expert verified
The coefficient of kinetic friction between the wheels and the runway is \(0.523\).

Step by step solution

01

Identify Known Quantities

By reading the exercise, we can define a list of known quantities:\nLinear speed of the plane: \(v = 50.0 \mathrm{~m/s}\)\nRadius of each wheel: \(r = 1.25 \mathrm{~m}\)\nMoment of inertia of each wheel: \(I = 110 \mathrm{~kg.m^2}\)\nForce supported by each wheel: \(F = 1.40 \times 10 ^ 4 \mathrm{~N}\)\nRolling without slipping time: \(t = 0.480 \mathrm{~s}\). The unknown quantity to find is the coefficient of kinetic friction \( \mu_k \).
02

Applying Physics Concepts

The angular speed \(蠅\) at which the wheels spin can be obtained using the equation \( v = 蠅r \). \nSolving this for \( 蠅 \), we get \( 蠅 = v/r = 50.0 \, \mathrm{m/s} / 1.25 \, \mathrm{m} = 40.0 \, \mathrm{rad/s} \).\nSince the wheels roll without slipping, the acceleration of the wheel is calculated by \( 伪 = 蠅/t = 40.0 \, \mathrm{rad/s} / 0.480 \, \mathrm{s} = 83.3 \, \mathrm{rad/s^2} \).\nThe Net torque can be calculated using the relation \( 蟿 = I伪 \). Thus, \( 蟿 = 110 \, \mathrm{kg.m^2} * 83.3 \, \mathrm{rad/s^2} = 9163 \, \mathrm{N.m} \).
03

Calculate friction and the coefficient of kinetic friction

Since \( 蟿 = Fr \), we can solve for \( F \), the frictional force. Thus \( F = 蟿 / r = 9163 \, \mathrm{N.m} / 1.25 \, \mathrm{m} = 7330 \, N\).\nUsing the formula of friction which states that the frictional force \( F \) is equivalent to the coefficient of kinetic friction \( 渭_k \) times the normal force \( N \), we can isolate \( 渭_k \) and solve for it: \( 渭_k = F / N = 7330 \, N / 1.4 \times 10^4 \, N = 0.523 \).

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Most popular questions from this chapter

A solid, uniform disk of radius \(0.250 \mathrm{~m}\) and mass \(55.0 \mathrm{~kg}\) rolls down a ramp of length \(4.50 \mathrm{~m}\) that makes an angle of \(15.0^{\circ}\) with the horizontal. The disk starts from rest from the top of the ramp. Find (a) the speed of the disk's center of mass when it reaches the bottom of the ramp and (b) the angular speed of the disk at the bottom of the ramp.

QIC S An Atwood's machine consists of blocks of masses \(m_{1}=\) \(10.0 \mathrm{~kg}\) and \(m_{2}=20.0 \mathrm{~kg}\) attached by a cord running over a pulley as in Figure P8.40. The pulley is a solid cylinder with mass \(M=\) \(8.00 \mathrm{~kg}\) and radius \(r=0.200 \mathrm{~m}\). The block of mass \(m_{2}\) is allowed to drop, and the cord turns the pulley without slipping. (a) Why must the tension \(T_{2}\) be greater than the tension \(T_{1}\) ? (b) Whatis the acceleration of the system, assuming the pulley axis is frictionless? (c) Find the tensions \(T_{1}\) and \(T_{2}\).

A large grinding wheel in the shape of a solid cylinder of radius \(0.330 \mathrm{~m}\) is free to rotate on a frictionless, vertical axle. A constant tangential force of \(250 \mathrm{~N}\) applied to its edge causes the wheel to have an angular acceleration of \(0.940 \mathrm{rad} / \mathrm{s}^{2}\). (a) What is the moment of inertia of the wheel? (b) What is the mass of the wheel? (c) If the wheel starts from rest, what is its angular velocity after \(5.00 \mathrm{~s}\) have elapsed, assuming the force is acting during that time?

QC A \(40.0-\mathrm{kg}\) child stands at one end of a \(70.0-\mathrm{kg}\) boat that is \(4.00 \mathrm{~m}\) long (Fig. P8.69). The boat is initially \(3.00 \mathrm{~m}\) from the pier. The child notices a turtle on a rock beyond the far end of the boat and proceeds to walk to that end to catch the turtle. (a) Neglecting friction between the boat and water, describe the motion of the system (child plus boat). (b) Where will the child be relative to the pier when he reaches the far end of the boat? (c) Will he catch the turtle? (Assume that he can reach out \(1.00 \mathrm{~m}\) from the end of the boat.)

A student sits on a rotating stool holding two \(3.0-\mathrm{kg}\) objects. When his arms are extended horizontally, the objects are \(1.0 \mathrm{~m}\) from the axis of rotation and he rotates with an angular speed of \(0.75 \mathrm{rad} / \mathrm{s}\). The moment of inertia of the student plus stool is \(3.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\) and is assumed to be constant. The student then pulls in the objects horizontally to \(0.30 \mathrm{~m}\) from the rotation axis. (a) Find the new angular speed of the student. (b) Find the kinetic energy of the student before and after the objects are pulled in.
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