Chapter 9 Rotational Dynamics

Rotational kinetic energy We consider a system of particles participating in rotational motion Kinetic energy of this system is Then

Moment of inertia From the previous slide Defining moment of inertia (rotational inertia) as We obtain for rotational kinetic energy

Moment of inertia: rigid body There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter, its distribution in the rigid object and also depends upon the location of the axis of rotation For a rigid body the sum is calculated over all the elements of the volume This sum can be calculated for different shapes and density distributions For a constant density and the rotation axis going through the center of mass the rotational inertia for common body shapes are well known

Moment of inertia: rigid body

Moment of inertia: rigid body The rotational inertia of a rigid body depends on the position and orientation of the axis of rotation relative to the body

Moment of inertia: rigid body Example: moment of inertia of a uniform ring The hoop is divided into a number of small segments, m1 …, which are equidistant from the axis

Chapter 9 Problem 39 A particle is located at each corner of an imaginary cube. Each edge of the cube is 0.25 m long, and each particle has a mass of 0.12 kg. What is the moment of inertia of these particles with respect to an axis that lies along one edge of the cube?

Torque The door is free to rotate about an axis through O Three factors that determine the effectiveness of the force in opening the door: 1) The magnitude of the force 2) The position of the application of the force 3) The angle at which the force is applied

Torque We apply a force at point P to a rigid body that is free to rotate about an axis passing through O Only the tangential component Ft = F sin φ of the force will be able to cause rotation

Torque The ability to rotate will also depend on how far from the rotation axis the force is applied Torque (turning action of a force): SI unit: N*m (don’t confuse with J)

τ = F r┴ Torque Torque: Moment arm (lever arm): r┴= r sinφ Torque can be redefined as: force times moment arm τ = F r┴

Torque Torque is the tendency of a force to rotate an object about some axis Torque is a vector The direction is perpendicular to the plane determined by the position vector and the force If the turning tendency of the force is CCW (CW), the torque will be positive (negative)

Torque When two or more torques are acting on an object, they are added as vectors The net torque is the sum of all the torques produced by all the forces If the net torque is zero, the object’s rate of rotation doesn’t change Forces cause accelerations, whereas torques cause angular accelerations

Newton’s Second Law for rotation Consider a particle rotating under the influence of a force For tangential components Similar derivation for rigid body

Newton’s Second Law for rotation

Chapter 9 Problem 34 A ceiling fan is turned on and a net torque of 1.8 N  m is applied to the blades. The blades have a total moment of inertia of 0.22 kg  m2. What is the angular acceleration of the blades?

Center of mass The force of gravity acting on an object must be considered In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point We wish to locate the point of application of the single force whose magnitude is equal to the weight of the object, and whose effect on the rotation is the same as all the individual particles This point is called the center of mass of the object

Center of mass In a certain reference frame we consider a system of particles, each of which can be described by a mass and a position vector For this system we can define a center of mass:

Center of mass of two particles A system consists of two particles on the x axis Then the center of mass is

Center of mass of a rigid body For a system of individual particles we have For a rigid body (continuous assembly of matter) the sum is calculated over all the elements of the volume

Angular momentum Angular momentum: SI unit: kg*m2/s Recall:

Conservation of angular momentum Newton’s Second Law for rotational motion If the net torque acting on a system is zero, then If no net external torque acts on a system of particles, the total angular momentum of the system is conserved (constant)

Conservation of angular momentum

Conservation of angular momentum

Chapter 9 Problem 63 A thin rod has a length of 0.25 m and rotates in a circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at one of its ends. The rod has an angular velocity of 0.32 rad/s and a moment of inertia of 1.1 × 10-3 kg  m2. A bug standing on the axis decides to crawl out to the other end of the rod. When the bug (mass = 4.2 × 10-3 kg) gets where it’s going, what is the angular velocity of the rod?

Equilibrium Equilibrium: Static equilibrium: Stable equilibrium: the body returns to the state of static equilibrium after having been displaced from that state Unstable equilibrium: the state of equilibrium is lost after a small force displaces the body

Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

Center of mass: stable equilibrium

The requirements of equilibrium For an object to be in equilibrium, we should have two requirements met Balance of forces: the vector sum of all the external forces that act on the body is zero Balance of torques: the vector sum of all the external torques that act on the body, measured about any possible point, is zero

Equilibrium: 2D case If an object can move only in 2D (xy plane) then the equilibrium requirements are simplified: Balance of forces: only the x- and y-components are considered Balance of torques: only the z-component is considered (the only one perpendicular to the xy plane)

Examples of static equilibrium

Examples of static equilibrium

Examples of static equilibrium

Examples of static equilibrium

Chapter 9 Problem 25 A 1220-N uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. A 1960-N crate hangs from the far end of the beam. Find (a) the magnitude of the tension in the wire and (b) the magnitudes of the horizontal and vertical components of the force that the wall exerts on the left end of the beam.

Indeterminate structures Indeterminate systems cannot be solved by a simple application of the equilibrium conditions In reality, physical objects are not absolutely rigid bodies Concept of elasticity is employed

Questions?