Newton’s laws
Read Tipler and Mosca (2004) chapter 4 (Newton’s
Laws) and chapter 5 (Application of Newton’s laws); Pelcovits and Farkas (2024)
chapter 3 (Newton’s laws).
Newton’s third law
Students usually remember Newton’s third law as
For every action, there is an equal and opposite reaction.
You should be able to describe the interaction of two objects or systems using Newton’s third law and a representation of paired forces exerted on each object or system. For example, if I push on the whiteboard and the whiteboard pushes back on me by the same amount, then we have satisfied Newton’s third law.
Newton’s third law describes the interaction of two objects or systems in terms of the paired forces that each exerts on the other: \[\begin{equation} \vec{F}_{A\ on\ B} = -\vec{F}_{B\ on\ A}. \end{equation}\]
Internal forces
Interactions between objects within a system (internal forces) do not influence the motion of a system’s center of mass. For example, a single bolt in the drink cart in a commercial airliner pulls on a nut; the nut pulls back on the bolt, but these are internal forces that do not change the motion of the airplane.
Tension in a rope
In AP Physics C problems, forces and the reactions to them are sometimes transmitted between different parts of the system using ropes, cables, or chains. The usual name for a force transmitted this way is tension (often given the variable \(T\)).
Tension is the macroscopic net result of forces that infinitesimal segments of a string, cable, chain, or similar system exert on each other in response to an external force. For AP purposes, an ideal string has negligible mass and does not stretch when under tension. The tension in an ideal string is the same at all points within the string. Real strings are non-ideal. In a real string with nonnegligible mass, tension may not be the same at all points within the string.
An ideal pulley is a pulley that has negligible mass and rotates about an axle through its center of mass with negligible friction.
Significance of Newton’s third law
I do not usually see a specific equation for Newton’s third law, but it shows up when we recognize that the force from one object to a second is equal and opposite to the force back from the second to the first. Newton’s third law is also seen in the tendency of mechanical and civil engineers to identify the force of the ground or an anchor onto a structure as the reaction force, usually given the variable \(R\).
Newton’s first law
Newton’s first law is often remembered by students as Avatar the Last Airbender fans will know this as
earthbending master Toph’s statement that ``… a rock does not throw
itself, Twinkletoes!’’.
An object at rest stays at rest, and an object in motion stays in motion, unless acted upon by an outside force.
This law is so important that an entire college course for engineers
called Staticse.g. MIT
2.001 Mechanics and Materials I
is based on it and it also pervades throughout other
courses in solid mechanics. It is commonly written in equation form as
\[\begin{equation}
\sum \vec{F} = 0.
\end{equation}\]
Newton’s first law describes the conditions under which a system’s velocity remains constant.
The net force on a system is the vector sum of all forces exerted on the system (\(\sum \vec{F}\)). Translational equilibrium is the configuration of forces such that the net force exerted on a system is zero. Newton’s first law states that if the net force exerted on a system is zero, the velocity of that system will remain constant, i.e. if it is at rest \(v=0\), it will stay at rest, if it is in motion at \(v\) it will stay in motion at the same \(v\).
Forces are vectors. As a result, forces may be unbalanced in one dimension or direction, but unbalanced in another. The system’s velocity will change only in the direction of the unbalanced force.
An inertial reference frame is one from which an observer would verify Newton’s first law of motion. Recall we said an inertial reference frame is one that is not accelerating. In an accelerating frame of reference, we may observe “pseudo-forces” owing to the acceleration of the frame.
Newton’s second law
Newton’s second law is usually remembered by students as \(F=ma\):
If there is a net force, it is equal to the time rate of change of momentum (\(\vec{p}=m\vec{v}\))
Momentum is a concept we will cover later, but was historically developed first. Since most of the systems we deal with in a first course in physics have constant mass, the time rate of change of momentum becomes \(\frac{d\vec{p}}{dt}=m\frac{d\vec{v}}{dt}=m\vec{a}\). It is also of great importance, engineers in college at the very least will take a course in dynamics which deals largely with \[\begin{equation} \sum \vec{F}=m\vec{a}. \end{equation}\]
Newton’s second law describes the conditions under which a system’s velocity changes.
Unbalanced forces, where \(\sum F\neq 0\), are a configuration of forces such that the net force exerted on a system is not equal to zero. Newton’s second law of motion states that the acceleration of a system’s center of mass has a magnitude proportional to the magnitude of the net force exerted on the system and is in the same direction as that net force. \[\begin{equation} \vec{a}_{sys} = \dfrac{\sum \vec{F}}{m_{sys}} = \dfrac{\vec{F}_{net}}{m_{sys}} \end{equation}\]
The velocity of a system’s center of mass will only change if a nonzero net external force is exerted on that system.
See also
- list here