Inertial vs Gravitational Mass (College Board AP® Physics 1: Algebra-Based)

Study Guide

Ann Howell

Written by: Ann Howell

Reviewed by: Caroline Carroll

Inertial versus gravitational mass

Inertial mass

  • Objects have inertial mass, or inertia

    • Inertia is a property that determines how much an object’s motion resists changes when interacting with another object

  • Newton's second law is used to calculate inertial mass

a with rightwards arrow on top subscript s y s end subscript space equals space F with rightwards arrow on top subscript n e t end subscript over m subscript s y s end subscript space rightwards arrow space F with rightwards arrow on top subscript n e t end subscript space equals space space m subscript s y s end subscript a with rightwards arrow on top subscript s y s end subscript

Measuring inertial mass

  • The inertial mass of an object can be determined by applying a force to accelerate the mass

    • This is commonly done by setting a mass oscillating on the end of a spring with a spring constant, k, and calculating the time period

Measuring inertial mass using a mass-spring system

A physics experiment setup with a clamp stand, ruler, spring, mass hanger holding a mass, a nail as a marker, and a stopwatch displaying 0.00 seconds.
Setting a mass on a spring oscillating to measure the time period of oscillating enables the inertial mass to be calculated
  • Set the mass oscillating and record the time for 10 oscillations using a stop watch

    • One oscillation involves the mass moving up to the top and back down to the bottom

  • Divide the total time for the 10 oscillations by 10 to obtain the time period

  • Using the equations for centripetal force, angular velocity and time period an equation for the spring constant, k, and the time period is obtained to calculate the inertial mass of the object

m subscript s y s end subscript space equals space fraction numerator k T squared over denominator 2 pi end fraction

Gravitational mass

  • Gravitational mass is related to the force of attraction between two systems with mass

  • Newton's law of gravitation is used to calculate the gravitational mass

open vertical bar stack F subscript g with rightwards arrow on top close vertical bar space equals space G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction

Measuring gravitational mass

  • The gravitational mass of an object can be determined by placing the object in a gravitational field and measuring the gravitational force, F subscript g, on the mass

    • This is commonly done by balancing the mass on a set of balance scales

    • The magnitude of the masses needed for the scales to be balanced (sitting horizontally on the pivot) is equal to the gravitational mass of the object

  • The balance scales below sit horizontally, showing that four single masses on the right equal the gravitational mass of the object on the left

Measuring gravitational mass using a balance scale

A balance scale with a large grey mass on the left pan and several smaller grey masses on the right pan, illustrating a weight comparison.
Balancing a mass on a balance scale to measure the magnitude of the mass needed to balance it gives the gravitational mass

Equivalence principle

  • Inertial mass and gravitational mass have been experimentally verified to be equivalent

    • Using the experimental methods above, the inertial and gravitational masses of an object are found to be the same

  • Algebraically, this means

F with rightwards arrow on top subscript n e t end subscript space equals space space m subscript s y s end subscript a with rightwards arrow on top subscript s y s end subscript space equals space open vertical bar stack F subscript g with rightwards arrow on top close vertical bar space equals space G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction

m subscript s y s end subscript a with rightwards arrow on top subscript s y s end subscript space equals space G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction

  • So, when m subscript s y s end subscript space equals space m subscript 1

a with rightwards arrow on top subscript s y s end subscript space equals space G m subscript 2 over r squared

  • Then the object accelerates at the same rate as the magnitude of the gravitational field strength

a with rightwards arrow on top subscript s y s end subscript equals space open vertical bar g with rightwards arrow on top close vertical bar space

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Ann Howell

Author: Ann Howell

Expertise: Physics Content Creator

Ann obtained her Maths and Physics degree from the University of Bath before completing her PGCE in Science and Maths teaching. She spent ten years teaching Maths and Physics to wonderful students from all around the world whilst living in China, Ethiopia and Nepal. Now based in beautiful Devon she is thrilled to be creating awesome Physics resources to make Physics more accessible and understandable for all students, no matter their schooling or background.

Caroline Carroll

Author: Caroline Carroll

Expertise: Physics Subject Lead

Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.