Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q. 71 (page 203)

If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.71, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equationz=ω22gr2

Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.

Short Answer

Expert verified

Shape of the parabola is described by the equation z=ω22gr2

Step by step solution

01

Given information

The figure shows the detail

02

Explanation

First draw Free Body diagram as below

From the diagram equate the vertical and radial force

Gravitational force, F = mg ………………..(1)

Radial Force Fradial=mrω2

Lets consider a very small element at radius dr and the height of the water above the surface is dh

The angle given as

tanϕ=mrω2mg=rω2g…………….…(2)tanϕ=dzdr………….....………(3)

From equation (2) and (3) we get

dzdr=rω2gdz=rω2gdr

Upon integration we get

z=ω22gr2

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

What is free-fall acceleration toward the sun at the distance of the earth’s orbit? Astronomical data are inside the back cover of the book .

A 100gbead slides along a frictionless wire with the parabolic shape y=2m-1x2.

a. Find an expression for ay, the vertical component of acceleration, in terms of x, vx, and ax. Hint: Use the basic definitions of velocity and acceleration.

b. Suppose the bead is released at some negative value ofx and has a speed of 2.3m/s as it passes through the lowest point of the parabola. What is the net force on the bead at this instant? Write your answer in component form

You’ve taken your neighbor’s young child to the carnival to ride the rides. She wants to ride The Rocket. Eight rocket-shaped cars hang by chains from the outside edge of a large steel disk. A vertical axle through the center of the ride turns the disk, causing the cars to revolve in a circle. You’ve just finished taking physics, so you decide to figure out the speed of the cars while you wait. You estimate that the disk is 5m in diameter and the chains are 6m long. The ride takes 10s to reach full speed, then the cars swing out until the chains are 20° from vertical. What is the cars’ speed?

Elm Street has a pronounced dip at the bottom of a steep hill before going back uphill on the other side. Your science teacher has asked everyone in the class to measure the radius of curvature of the dip. Some of your classmates are using surveying equipment, but you decide to base your measurement on what you’ve learned in physics. To do so, you sit on a spring scale, drive through the dip at different speeds, and for each speed record the scale’s reading as you pass through the bottom of the dip. Your data are as follows:

Speed m/sec

Scale Reading N

5

599

10

625

15

674

20

756

25

834

In the Bohr model of the hydrogen atom, an electron (mass m=9.1×10-31kg)orbits a proton at a distance of 5.3×10-11m. The proton pulls on the electron with an electric force of8.2×10-8N. How many revolutions per second does the electron make?
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Fields in Physics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Linear Momentum

Read Explanation

Electricity and Magnetism

Read Explanation

Atoms and Radioactivity

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.

Sign-up for free