5. Impulse, Torque, & Angular Momentum for a System of Particles

MIT OpenCourseWare・2 minutes read

MIT OpenCourseWare offers educational resources for free, supported by donations. The lecture covers topics such as tangent and normal unit vectors, impulse, linear momentum, and angular momentum.

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  • MIT OpenCourseWare provides free educational resources supported by donations, covering technical topics like tangent and normal unit vectors, impulse, linear momentum, and angular momentum.
  • The lecture delves into detailed calculations involving acceleration, radius of curvature, linear impulse, and momentum, emphasizing the importance of understanding torque and forces in systems with rotating components like carnival rides, showcasing real-world applications of theoretical concepts.

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Summary

00:00

"Free MIT lecture on tangent vectors"

  • MIT OpenCourseWare offers educational resources for free, supported by donations.
  • The technical topic discussed includes tangent and normal unit vectors.
  • The lecture covers a review of impulse, linear momentum, and angular momentum.
  • Tangent and normal coordinates are explained using a driving example on a curve.
  • Velocity is described using tangent and normal unit vectors.
  • Acceleration is calculated by taking derivatives and considering centripetal acceleration.
  • The radius of curvature is determined using mathematical formulas.
  • An example with a sine function is used to calculate acceleration and radius of curvature.
  • The acceleration is found to be 3.51 meters per second squared, pointing inward.
  • The lecture transitions to linear impulse and momentum, discussing Newton's Second Law and the law of conservation of momentum.

19:25

Impulse-Momentum Principles in Physics Systems

  • The object in the x direction starts at v1 with x=0 and 0 velocity when released.
  • The integral of k dt is kt, leading to mv2 in the x direction when t=3 seconds.
  • The problem simplifies to 3g sine theta minus mu cosine theta.
  • The process involves starting with a vector equation and applying it in any vector component direction.
  • Forces are integrated over time, and the impulse-momentum formula is applied to find the answer.
  • Conservation of momentum applies when there are no forces present.
  • For systems of particles, the total mass times the velocity of the center of mass is crucial.
  • The time derivative of momentum equals external forces, leading to the acceleration of the center of mass.
  • The change in linear momentum of the system is equal to the integral of forces over time.
  • The reaction force on a cannon when firing a shot is equal and opposite to the force on the shot, calculated using impulse and momentum principles.

39:09

Calculating Torque and Angular Momentum in Systems

  • To calculate angular momentum with respect to another point, use the formula hB with respect to A equals rB with respect to A cross PB with respect to o.
  • Momentum is always calculated with respect to the inertial frame, emphasizing the importance of specifying the frame.
  • The torque on a system around a particle B with respect to A is the time rate of change of hB/A with respect to t, plus the velocity of A with respect to o cross PB with respect to o.
  • An example scenario where this calculation is useful is designing a motor to rotate an arm with a mass attached.
  • The sum of forces at B gives the time rate of change of momentum at B with respect to o, considering external forces acting on the particle.
  • The torque with respect to A is the time derivative of rB/A cross PB/o minus the derivative of rB/A cross PB/o.
  • The torque can be simplified in cases where the velocity of A is 0 or parallel to the direction of momentum, especially when A is at the center of mass.
  • For rigid bodies, the torque with respect to a point A is the time rate of change of angular momentum with respect to A plus the velocity of A with respect to o cross PB/o.
  • The torque with respect to the center of mass is the time rate of change of angular momentum with respect to the center of mass.
  • The formulas derived are crucial for understanding and calculating torques and forces in systems with rotating components, such as carnival rides with moving arms.

01:04:34

Calculating Torque and Acceleration in Dynamics

  • The first piece to obtain is the angular momentum, denoted as h B/o.
  • To determine the torque at point B, only the first term of the formula is necessary since the velocity of the point is constant.
  • The torque required to move the particle at B with respect to o is the time rate of change of h B with respect to o.
  • The torque expression consists of two terms: one from the Coriolis force and the other from the Eulerian force.
  • By calculating the Coriolis force and torque, it is possible to determine the acceleration of the system.
  • The acceleration due to the Coriolis force is approximately 2.4 meters per second squared, equivalent to about a quarter of a g.
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