Chapter 6: Q17P (page 285)

Expand the triple product for $a = ω \times (ω \times r)$ given in the discussion of Figure 3.8. If is perpendicular to $w$ (Problem 16), show that $a = ‐ ω^{2} r$ , and so find the elementary result that the acceleration is toward the center of the circle and of magnitude $\frac{v^{2}}{r}$ .

Short Answer

Expert verified

The acceleration $a = ω (ω . r) ‐ r (ω^{2})$

$|a| = \frac{v^{2}}{r}$ Is proved.

Step by step solution

Given condition

The given condition is, $a = w (w . r) ‐ r (w^{2})$ .

Concept Used

The value of a triple vector product is a linear combination of the two vectors in the parenthesis; the coefficient of each vector is the dot product of the other two; the middle vector in the triple product always has the positive sign.

Determine the Acceleration

Using the identity of the vector triple product:

$A \times (B \times C) = B (C . A) ‐ C (B . A) a = ω (ω . r) ‐ r (ω . ω) a = ω (ω . r) ‐ r (ω^{2})$ .

The acceleration $a = ω (ω . r) ‐ r (ω^{2})$

Determine the equation

The perpendicular vectors are given as follows,

From $ω$ is perpendicular to $r$ , $(ω . r) = 0$

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Expand the triple product for $a = ω \times (ω \times r)$ given in the discussion of Figure 3.8. If is perpendicular to $w$ (Problem 16), show that $a = ‐ ω^{2} r$ , and so find the elementary result that the acceleration is toward the center of the circle and of magnitude $\frac{v^{2}}{r}$ .

Short Answer

Step by step solution

Given condition

Concept Used

Determine the Acceleration

Determine the equation

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Expand the triple product for a=ω×(ω×r) given in the discussion of Figure 3.8. If is perpendicular to w(Problem 16), show that a=‐ω2r, and so find the elementary result that the acceleration is toward the center of the circle and of magnitude v2r.

Short Answer

Step by step solution

Given condition

Concept Used

Determine the Acceleration

Determine the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Waves Physics

Dynamics

Electricity and Magnetism

Linear Momentum

Further Mechanics and Thermal Physics

Classical Mechanics

Study anywhere. Anytime. Across all devices.

Expand the triple product for $a = ω \times (ω \times r)$ given in the discussion of Figure 3.8. If is perpendicular to $w$ (Problem 16), show that $a = ‐ ω^{2} r$ , and so find the elementary result that the acceleration is toward the center of the circle and of magnitude $\frac{v^{2}}{r}$ .