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Ch 08: Dynamics II: Motion in a Plane
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 08: Dynamics II: Motion in a PlaneProblem 39
Chapter 8, Problem 39

A 75 kg man weighs himself at the north pole and at the equator. Which scale reading is higher? By how much? Assume the earth is spherical.

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Step 1: Understand the problem. The man weighs himself at two locations: the north pole and the equator. The scale reading measures the normal force exerted by the ground on the man, which is affected by gravitational force and centripetal force due to Earth's rotation. We need to determine which reading is higher and by how much.
Step 2: Analyze the forces acting on the man. At the north pole, the gravitational force is the only force acting on him since there is no rotational motion. At the equator, the gravitational force is reduced by the centripetal force due to Earth's rotation. The centripetal force is given by \( F_c = m \cdot \omega^2 \cdot r \), where \( m \) is the mass of the man, \( \omega \) is the angular velocity of Earth, and \( r \) is the radius of Earth.
Step 3: Calculate the gravitational force at both locations. The gravitational force is given by \( F_g = m \cdot g \), where \( g \) is the acceleration due to gravity. At the north pole, \( g \) is unaffected by rotation, while at the equator, \( g \) is slightly reduced due to the centripetal force.
Step 4: Determine the scale reading at each location. At the north pole, the scale reading equals the gravitational force \( F_g \). At the equator, the scale reading equals \( F_g - F_c \), where \( F_c \) is the centripetal force. Subtract the centripetal force from the gravitational force to find the scale reading at the equator.
Step 5: Compare the scale readings. The reading at the north pole will be higher because the centripetal force reduces the effective weight at the equator. To find the difference, subtract the scale reading at the equator from the scale reading at the north pole.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Weight and Gravitational Force

Weight is the force exerted on an object due to gravity, calculated as the product of mass and gravitational acceleration (W = mg). The gravitational force varies slightly depending on location due to the Earth's shape and rotation, affecting scale readings.
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Centrifugal Force

Centrifugal force is an apparent force that acts outward on a mass moving in a circular path, resulting from the rotation of the Earth. At the equator, this force counteracts gravitational pull slightly more than at the poles, leading to a lower weight reading on a scale.
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Variation in Gravitational Acceleration

Gravitational acceleration varies with latitude due to the Earth's rotation and its equatorial bulge. At the poles, gravitational acceleration is approximately 9.83 m/s², while at the equator, it is about 9.78 m/s², causing a difference in weight measurements between these two locations.
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