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Newton's second law
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Definition/Summary
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Net force = rate of change of momentum
Net impulse = change of momentum
These are vector equations, so they apply to each direction individually |
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Equations
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If m is constant:
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Recent forum threads on Newton's second law
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Breakdown
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Physics
> Classical Mechanics
>> Newtonian Dynamics
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Extended explanation
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Net force = mass times acceleration:
This only applies when the mass is constant:
For example, it does not apply to a rocket, whose mass is continually reduced by the burning of fuel.
Standard (non-impulsive) version:
Newton's second law states that the total vector sum of the forces on a body is equal to the rate of change of its momentum. Component version:
Force and momentum are vectors, so Newton's second law is a vector equation.
That means that it applies to the components in any direction, and so the law can also be written:
In any direction, the total sum of the components of the forces on a body in that direction is equal to the rate of change of the component of its momentum in that direction. Torque:
The vector nature of the equation also means that its cross-product with any fixed vector gives another vector equation:
Net torque on a point particle = rate of change of angular momentum:
Combining this with the strong version of Newton's third law (action is equal and opposite to reaction, and in the same line) gives:
Net torque on a rigid body = rate of change of angular momentum:
This is also a vector equation, of course, and so we may take one component at a time, to give the scalar equation:
Net torque about any axis = rate of change of angular momentum about that axis. Impulsive version:
Impulse is the integral of force times time.
By comparison, work done is the integral of force times distance.
Sometimes (for example, when a bat hits a ball), the force changes quickly, and it is difficult to measure it, but it is easy to measure the overall effect of the force. The impulse is that overall effect.
Newton's second law states that the total vector sum of the impulses on a body is equal to the change of its momentum. Collisions, and conservation of momentum:
A collision or explosion is impulsive, and there are no external impulsive forces in a collision or explosion (in particular, the sum of the external forces is zero  ), so the impulsive version of the law states:
Total momentum immediately before = total momentum immediately after:
In other words: conservation of momentum applies to all collisions or explosions.
By comparison, conservation of energy applies only to elastic collisions. Most collisions are not elastic.
Static problems:
When nothing moves, the momentum is constant, and so Newton's second law states:
The total sum of the forces on a body in equilibrium is zero.
This applies separately to each body in a system, or to the external forces on any combination of bodies.
When there are only three forces, a vector triangle may be used.
Sliding problems (normal components):
When one body slides against another along a common flat surface, the acceleration normal (perpendicular) to that surface is zero.
If the surface is curved, the acceleration normal to the surface is not zero, but is centripetal towards the instantaneous centre of curvature of the surface (note: this is a matter of geometry, not physics ).
Accordingly, Newton's second law for normal components states:
The total sum of the components of the normal forces on a sliding body on a flat surface is zero, and on a curved surface equals the mass times the centripetal acceleration. |
Commentary
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