Rotational Motion

• We say that an axis of rotation is a symmetry axis if the object can be rotated along this axis such that the positions before and after the rotation are indistinguishable.

Tangential Kinematics §

• A point mass moves in uniform circular motion if it moves in a circle at constant speed.

  • In this scenario, there is no change of acceleration parallel to the path. Thus, acceleration is perpendicular to the path and directed inward.

  • Let be the magnitude of acceleration of an object in uniform circular motion. If the object travels in radius and speed , we have

  • Let be the period of motion. The speed is given as

• When we have non-uniform circular motion, a component of acceleration is tangent to the instantaneous velocity. This vector is given by and is defined as

  The component perpendicular to the motion is still given by above.

• Objects in circular motion experience Centripetal force.

Orbit Kinematics §

• For the following, we assume that any bodies are idealized rigid bodies
• The angular coordinate is the angle between the object and an axis of rotation. We assume that the axis of rotation is fixed with respect to an inertial frame of reference
• The angular velocity is defined as
  • At any instant, every part of a rotating rigid body has the same angular velocity. This is because all parts of the system must travel the same angle.
  • The angular velocity vector is directed along the axis of rotation, perpendicular to the plane of rotation following the right hand rule.
  • The tangential velocity is related to the angular velocity as follows
    Where represents the cross product and is the position vector relative to the axis of rotation.
  • The tangential speed is related to the angular speed by
• The angular acceleration is defined as

  • The angular acceleration is related to the tangential acceleration as follows

    Where represents the cross product and is the position vector relative to the axis of rotation.

  • The magnitudes of the angular and tangential acceleration is related by

  • The centripetal component of the acceleration can then be derived as

  • When the angular acceleration is constant, the kinematic equations still hold except with instead of respectively.

Rotational Dynamics §

• The mass analogue in this case is the moment of inertia.

  • The more particles are far away from the axis of rotation, the higher the moment of inertia.
  • The rotational kinetic energy of the rigid body is given by
  • The greater the moment of inertia, the greater the kinetic energy of the rigid body rotating.
  • A higher rotational inertia means that it is more difficult to change the system’s angular acceleration (analogous to regular inertia).

• If the rotation axis of a rigid body is not a symmetry axis, does not lie along the rotation axis. A net torque would be required to maintain rotation

Torque §

• Let be a force. The torque / moment of the force applied to a rotating body is defined as

  Where is the position vector of the force along the moment arm defined with respect to a point from which torque is being measured

  If is the component of the force perpendicular to following the right hand rule, then

  • Torque is the rotational analogue of Force.
  • Notation: By convention, we assume that unless otherwise stated, the torque is measured from the same axis as the angular displacement, velocity and acceleration

• Varignon’s Theorem states that the net torque is also the sum of the individual torques due to each force. That is

• A rotational analogue of Newton’s Second Law states that

  Where the angular acceleration and torque are relative to the same axis.

  Note this only applies for rigid bodies where angular acceleration is the same for all parts of the system.

  • The torque on each particle is due to the net force on that particle.
  • The torques for each pair of particle are equal and opposite because of Newton’s third law.
  • All internal torques sum to so the rotational analogue gives the sum of the external torques.
  • This applies even for a moving axis of rotation as long as.
    • The axis through the center of mass is an axis of symmetry.
    • The axis must not change direction,

• An object’s center of gravity is the point on a body where the torque due to Gravity vanishes.

  • The torque due to gravity is given by
  • If has the same value at all points on an object, the center of gravity is identical to the center of mass.
  • In general, however, the center of gravity need not be at the center of mass
  • In most cases near the earth’s surface, we can approximate that the center of gravity equals the center of mass.
  • An object in rotational equilibrium and acted on by gravity supported by a single point has a center of gravity along an axis that passes through extending in the direction of gravity.
  • An object with multiple supports will have its center of gravity somewhere within the area bounded by the supports
    • The lower the center of gravity and the larger the area of support, the harder to overturn an object

Work and Energy §

• For a system that is both moving and rotating, the kinetic energy is given as the sum of the kinetic energy using the linear and rotational formulae for kinetic energy.

  We use a moment of inertia running through the center of mass

  • Intuition: The additive relationship derives from the velocity having a translational and rotational component which can be separated. Remember to treat
  • In order for an object to roll without slipping, the point on the object that contacts the surface must instantaneously be at rest so that it does not slip. This is achieved when
    Where is the radius of the wheel.
    • For a rigid body that rolls without slipping, the kinetic energy of rotation depends on the shape of the rigid body.

• The Work done by a torque is given using the following integral

  Where torque and angular displacement is measured in the same direction for rotation about a fixed axis through the center of mass.

  • The Work-Energy theorem still applies
  • Power can also be derived similarly by using

Momentum §

• The angular momentum is the analogue for momentum. Assuming it is taken relative to the origin . It is defined as follows for situations when the axis of rotation is a symmetry axis

• It is related to linear momentum by

  Where is the position relative to and assuming we are in an inertial frame of reference.

• The angular momentum is related to the torque in a similar manner as linear momentum and force

• Law of Conservation of Angular Momentum. When the net external torque acting on a system is , the total angular momentum of the system is constant. That is for a system with particles

  Where is some constant.

  • As with collisions, internal forces can transfer angular momentum but can’t change the total angular momentum of the system.

Precession §

• A precession is the movement of the axis around another axis due to torque acting to change the direction of the first axis.
• Precession is simply the rotational analog of uniform circular motion (assuming it is torque-free). The angular velocity changes orientation with time which induces a varying moment of inertia.
  • In this case, the velocity of each axis varies inversely to the axis’ moment of inertia.

Links §

• University Physics with Modern Physics 15th Edition By Freedman - Ch. 9 - 10