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The Physics Hypertextbook © 1998–2013 Glenn Elert |
No condition is permanent.
Taste and compare …
| translational | rotational | ||
|---|---|---|---|
| 1st | An object at rest tends to remain at rest and an object in motion tends to continue moving with constant velocity unless compelled by a net external force to act otherwise. | An object at rest tends to remain at rest and an object in rotation tends to continue rotating with constant angular velocity unless compelled by a net external torque to act otherwise. | |
| 2nd | Acceleration is directly proportional to net force and inversely proportional to mass or ∑F = m a |
Rotational acceleration is directly proportional to net torque and inversely proportional to moment of inertia or ∑τ = I α |
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| 3rd | For every action there is an equal and opposite reaction. (Here action and reaction refer to forces.) | For every action there is an equal and opposite reaction. (Here action and reaction refer to torques.) |
| concept | translation | connection | rotation | |||
|---|---|---|---|---|---|---|
| cause of acceleration | ∑ F | τ = | r × F | ∑ τ | ||
| resistance to acceleration | m | I = | ∑ ri2mi = ∫ r2 dm | I | ||
| newton's second law | ∑ F = | m a | ∑ τ = | I α | ||
Start with the equation for Newton's second law. Cross radius into both sides and then play around with the algebra. (This requires some unusual identities that I've seen before, but never really learned.)
| ∑F = | m a |
| ∑r × F = | m r × a = m r × (α × r) |
| ∑r × F = | m α(r · r) − r(r · α) = m α r2 − 0 = mr2 α |
| ∑τ = | I α |
In the end, we get an analogous formula for Newton's second law of rotation with two new quantities: torque, the rotational equivalent to force
τ = r × F
and moment of inertia, the rotational equivalent to mass
| I = ∑ ri2 mi = | ⌠ ⌡ |
r2 dm |
More …
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The Physics Hypertextbook © 1998–2013 Glenn Elert |
No condition is permanent.