Turn on thread page Beta

Volume and surface area of semi-circular groove. watch

    • Thread Starter
    Offline

    0
    ReputationRep:
    I am looking at Stroud Engineering Mathematics Chapter 19 question 16.

    "A groove, semi-circular in section and 1cm deep, is turned in a solid cylindrical shaft of diameter 6cm. Find the volume of material removed and the surface area of the groove."

    So from frame 39 in the chapter, Pappus says:-

    "If an arc of a plane curve rotates about an axis in its place, the area of the surface generated is equal to the length of the line multiplied by the distance travelled by its centroid"

    and

    "If a plane figure rotates about an axis in its plane, the volume generated is equal to the area of the figure multiplied by the distance travelled by its centroid."

    So I am looking for the volume of material removed and think its something to do with the second part.

    The area of the semi-circle is pi/2 (as r is 1).

    The centroid of a semi-circle from the centre is 4/3pi. My semi circle is has the centre facing upwards and is on the surface of a 6cm diameter cylinder. So the distance of the centroid from the origin is 3-4/3pi.

    The centroid therefore travels 2pi(3-4/3pi)

    And so the volume removed is pi/2*2pi*(3-4/3pi) = 25.42 the answer in the back. So I am happy.

    I then go to the surface area of the groove from the first pappus statement must be the length of the arc * distance travelled by its centroid.

    So the length of my arc is pi (as r = 1). The distance travelled by the centroid is surely the same as in the first part and so I should have pi*2pi*(3-4/3pi) = 50.84, which isn't the answer in the book.

    Any idea what I am missing?

    Thanks
    Anthony
    • Study Helper
    Offline

    15
    Study Helper
    (Original post by antlee)
    Any idea what I am missing?
    The centroid of a solid semicircle (an area) is different to the centroid of the arc of a semicircle (a line).
    • Thread Starter
    Offline

    0
    ReputationRep:
    Thanks, I see.

    I looked up the centroid of the semi-circular arc (line) and it's 2/pi. And I pop that into what I was doing before and get the answer in the book.

    What I can't quickly find on web is the derivation of this.

    I derived the centroid of the shape myself

    Area*y(bar) = 1/2 int (y^2) dx between -1 and 1 (radius is 1 in this question) and this game me 4/3pi

    Can you point me in the direction of how to derive 2/pi or suggest what I integrate and divide by the length of the line.

    Thanks
    Anthony
    • Study Helper
    Offline

    15
    Study Helper
    (Original post by antlee)

    Can you point me in the direction of how to derive 2/pi or suggest what I integrate and divide by the length of the line.

    Thanks
    Anthony
    I'd suggest using polar co-ordinates.


    \pi r\bar{x}=\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} r\times r\cos\theta\;\mathrm{d\theta}

    r is obviously 1, but I've put it in as r for clarity of the calculation.
 
 
 

University open days

  1. University of Bradford
    University-wide Postgraduate
    Wed, 25 Jul '18
  2. University of Buckingham
    Psychology Taster Tutorial Undergraduate
    Wed, 25 Jul '18
  3. Bournemouth University
    Clearing Campus Visit Undergraduate
    Wed, 1 Aug '18
Poll
How are you feeling in the run-up to Results Day 2018?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.