Chapter 1: Q10. (page 18)

What, roughly, is the percent uncertainty in the volume of a spherical beach ball of radius $r = 0.84 \pm 0.04 m$ ?

Short Answer

Expert verified

The correct answer of percent uncertainty is 10%.

Step by step solution

Step 1. Writing the given data

Volume is defined as the amount of space that a particular object or body occupies in the space.

The radius of the spherical beach ball, $r = 0.84 \pm 0.04 m$ .

Writing the formula of the volume of a sphere

The formula of the volume of a sphere is

$V = \frac{4}{3} π r^{3} \dots (i)$

Calculating the volume of the spherical beach ball

From equation (i), the volume is

$V = \frac{4}{3} π {(0.84)}^{3}$ .

Thus, the final value of the multiplication is0.6%.

Step 4. Writing the formula for a small change in volume

Differentiating equation (i) w.r.t to radius,

$\frac{d V}{d r} = \frac{4}{3} π 3 r^{2} d r$ .

The formula of the small change in the volume of a sphere is

role="math" localid="1643623569468" $d V = 4 π r^{2} d r \dots (i i)$ .

Step 5. Calculating the value of a small change in volume

From equation (ii), the small change in the volume is

$d V = 4 π {(0.84)}^{2} (0.04)$ .

Step 6. Writing the formula of percent uncertainty

The formula of percent uncertainty is

$V' = \frac{d V}{V} \times 100 % \dots (i i i)$ .

Step 7. Calculating the value of percent uncertainty

From equation (iii), the value of percent uncertainty is

$\begin{matrix} V' = \frac{4 π {(0.84)}^{2} (0.04)}{\frac{4}{3} π {(0.84)}^{3}} \times 100 % \\ \approx 10 % \end{matrix}$

Thus, the final value is 10%.

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What, roughly, is the percent uncertainty in the volume of a spherical beach ball of radius $r = 0.84 \pm 0.04 m$ ?

Short Answer

Step by step solution

Step 1. Writing the given data

Writing the formula of the volume of a sphere

Calculating the volume of the spherical beach ball

Step 4. Writing the formula for a small change in volume

Step 5. Calculating the value of a small change in volume

Step 6. Writing the formula of percent uncertainty

Step 7. Calculating the value of percent uncertainty

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What, roughly, is the percent uncertainty in the volume of a spherical beach ball of radiusr=0.84±0.04m?

Short Answer

Step by step solution

Step 1. Writing the given data

Writing the formula of the volume of a sphere

Calculating the volume of the spherical beach ball

Step 4. Writing the formula for a small change in volume

Step 5. Calculating the value of a small change in volume

Step 6. Writing the formula of percent uncertainty

Step 7. Calculating the value of percent uncertainty

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Fluids

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Conservation of Energy and Momentum

Fundamentals of Physics

Waves Physics

Study anywhere. Anytime. Across all devices.

What, roughly, is the percent uncertainty in the volume of a spherical beach ball of radius $r = 0.84 \pm 0.04 m$ ?