Q. 37 Prove that the normal component ... [FREE SOLUTION]

Chapter 11: Q. 37 (page 899)

Prove that the normal component of acceleration is $0$ at a point of inflection on the graph of a twice-differentiable function $y = f (x)$ .

Short Answer

Expert verified

Let $r (t) = <t, f (t), 0> .$ Then,

role="math" localid="1649616080437" $v (t) = <1, f' (t), 0>, a (t) = <0, f'' (t), 0>$

We have $v (t) 脳 a (t) = <0, 0, f'' (t)>$ . At a point of inflection we have $a_{T} = \frac{||v (t) 脳 a (t)||}{||v (t)||} a_{T} = 0$ .

Step by step solution

Step 1. Given information.

Consider the given question,

A twice-differentiable function is $y = f (x)$ .

Step 2. Consider each inflection point f''t=0.

We know that the each inflection point $f'' (t) = 0$ .

Assume $r (t) = <t, f (t), 0> v (t) = r' (t) d t v (t) = <t, f' (t), 0> a (t) = v' (t) d t a (t) = <0, f'' (t), 0>$

Consider $v (t) 脳 a (t)$ . Then,

$v (t) 脳 a (t) = |\begin{matrix} i & j & k \\ 1 & f' (t) & 0 \\ 0 & f'' (t) & 0 \end{matrix}| = i (0) - j (0) + k (f'' (t)) = <0, 0, f'' (t)>$

Step 3. Determine the normal component of acceleration.

Consider the given question,

$||v (t) 脳 a (t)|| = \sqrt{{(f'' (t))}^{2}} = f'' (t) = 0$

At inflection point $f'' (t) = 0$ .

The normal component of acceleration is given below,

$a_{N} = \frac{||v (t) 脳 a (t)||}{||v||} = \frac{0}{||v||} = 0$

Thus, the normal component of acceleration is $0$ at a point of inflection of the graph of $f (x)$ which is twice differentiable.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Prove that the normal component of acceleration is $0$ at a point of inflection on the graph of a twice-differentiable function $y = f (x)$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider each inflection point f''t=0.

Step 3. Determine the normal component of acceleration.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Decision Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Prove that the normal component of acceleration is 0at a point of inflection on the graph of a twice-differentiable functiony=f(x).

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider each inflection point f''t=0.

Step 3. Determine the normal component of acceleration.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Decision Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Prove that the normal component of acceleration is $0$ at a point of inflection on the graph of a twice-differentiable function $y = f (x)$ .