Ch 01: Units & VectorsWorksheetSee all chapters
All Chapters
Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
Ch 05: Friction, Inclines, Systems
Ch 06: Centripetal Forces & Gravitation
Ch 07: Work & Energy
Ch 08: Conservation of Energy
Ch 09: Momentum & Impulse
Ch 10: Rotational Kinematics
Ch 11: Rotational Inertia & Energy
Ch 12: Torque & Rotational Dynamics
Ch 13: Rotational Equilibrium
Ch 14: Angular Momentum
Ch 15: Periodic Motion (NEW)
Ch 15: Periodic Motion (Oscillations)
Ch 16: Waves & Sound
Ch 17: Fluid Mechanics
Ch 18: Heat and Temperature
Ch 19: Kinetic Theory of Ideal Gasses
Ch 20: The First Law of Thermodynamics
Ch 21: The Second Law of Thermodynamics
Ch 22: Electric Force & Field; Gauss' Law
Ch 23: Electric Potential
Ch 24: Capacitors & Dielectrics
Ch 25: Resistors & DC Circuits
Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
Ch 29: Alternating Current
Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
Ch 36: Atomic Structure
Ch 37: Nuclear Physics
Ch 38: Quantum Mechanics

Solution: The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to 53 cm, and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of 9 cubits and a diameter of 2 cubits. For the stated range, what are the lower value and the upper value, respectively, for (a) the cylinder’s length in meters, (b) the cylinder’s volume in cubic meters? (c) What is the percent uncertainty in the volume?

Solution: The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to 53 cm, and suppose that anci

Problem

The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to 53 cm, and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of 9 cubits and a diameter of 2 cubits. For the stated range, what are the lower value and the upper value, respectively, for (a) the cylinder’s length in meters, (b) the cylinder’s volume in cubic meters? (c) What is the percent uncertainty in the volume?

Solution

For operations with uncertainty, we’ll follow two different rules depending on whether we're adding/subtracting or multiplying/dividing:

  • When measurements are added or subtracted, sum the absolute or relative uncertainty—the result is the same. 
  • When measurements are multiplied or divided, sum the relative uncertainties.

So anytime you square a measurement, add the uncertainty twice.

To convert between absolute uncertainty and percent uncertainty, we’ll use this formula (= measurement, Δ= absolute uncertainty):

m±u=m±um×100%

This problem gives us the maximum and minimum values for the length of a cubit. For parts (a) and (b) of the problem, we just need to convert the units from centimeters to meters, write our equation, and plug in the max and min. Part (c) is a little more complicated: we’ll need the rule for multiplying measurements with uncertainty.

We’ll assume that we know the length and diameter of the cylinder to 0.1 cubit—that is, = 9.0 cubits and d = 2.0 cubits.

(a) For this part, we just multiply h = 9.0 cubits by the minimum and maximum length of a cubit:

hmin=9(43 cm)10-2 m1 cm=9(0.43 m)=3.87 m  and  hmax=9(53 cm)10-2 m1 cm=9(0.53 m)=4.87 m

Solution BlurView Complete Written Solution