Mechanics
Essential formulas and concepts in mechanics
Projectile Range (level ground)
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Newton’s Second Law
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Fundamental law relating force, mass, and acceleration. It implies acceleration is proportional to net force and inversely proportional to mass.
Weight (Force due to Gravity)
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Weight is the gravitational force exerted on a mass by Earth (or another celestial body). It differs from mass; mass is constant, weight depends on local \(g\).
Momentum
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Impulse
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Momentum is the quantity of motion (mass times velocity). Impulse is the effect of a force acting over time, equal to the change in momentum. These are illustrated by collisions: e.g., a longer impact time reduces force for the same momentum change (crumple zones in cars).
Work (Constant Force)
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Power
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Power is the rate of energy transfer or work done per second (measured in Watts). For example, 1 watt = 1 joule/second. The \(Fv\) form is useful for constant forces (e.g., power required to lift an object at constant speed). Often visualized with mechanical engines or electrical power calculations.
Newton’s Law of Universal Gravitation
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Every mass exerts an attractive force on every other mass. This inverse-square law explains planetary orbits, moon’s gravity, etc.
Gravitational Potential Energy (two masses)
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- \(Same m_1, m_2, r, G as above. The negative sign indicates that the potential energy is zero at infinite separation and becomes more negative as masses come closer (bound state).\)
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Energy associated with two masses in a gravitational field. As \(r\) decreases, \(U\) decreases (becomes more negative), meaning work would be required to separate them. Often visualized as a well or curve approaching 0 at infinity and dipping down as \(r\) decreases.
Moment of Inertia (for discrete masses)
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- \(m_i are point masses in the object, r_i is each mass’s distance to the axis of rotation. I depends on the axis chosen.\)
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Moment of inertia is rotational inertia, representing how mass is distributed relative to the axis. It appears in rotational dynamics (\(I\) plays the role of \(m\)). For example, a solid disk vs. a hoop of equal mass have different \(I\) about the center (hoop has more mass at perimeter, larger \(I\)).
Rotational Dynamics
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This is the rotational analogue of \(F=ma\). It states that net torque produces angular acceleration proportional to the moment of inertia. For example, it requires more torque to spin up a heavier or more extended object at the same rate.