Angular impulse
The angular impulse in physics is the product of the average of a briefly applied force F times the radius of a rigid body R times the brief time period δt:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F}\times{R}\times\delta{t}= I \times(\omega_f - \omega_i)}
where the final angular velocity is just after the impulse, and the initial angular velocity is just before the impulse.
Example
Suppose a rotating cylinder of radius R2 is joined with a stationary cylinder of radius R1. Both feel an impulsive force until the continue rotating without slippage at different angular velocities. The above formula permits calculation of the respective angular velocities as follows:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F_1}\times\delta{t_1}= -\mathbf{F_2}\times\delta{t_2}}
Thus:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{I_1\times(\omega_{f1} - \omega_{i1})}{R_1} = - \frac{I_2\times(\omega_{f2} - \omega_{i2})}{R_2}}
But:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{i2} = 0}
and:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{f1} = \frac{\omega_{f2}\times{R_2}}{R_1}}
and thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{f2}} can be solved in terms of the initial angular velocity of the rotating cylinder Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_{i1}} .