Section 3.5 – Curvilinear Motion We now consider particles moving along plane curvilinear paths. We use the same coordinate systems that we used to describe the kinematics of a particle undergoing plane curvilinear motion: Rectangular Coordinates: Normal & Tangential Coordinates: Polar Coordinates:
Steps to Follow Identify the Motion Choose appropriate Coordinate System Draw FBD Obtain net force for each component Note: once you assign a coordinate system you must use expressions for both the forces and the accelerations that are consistent with the coordinate system.
Example Problem (n-t Coords)
Practice Problem (n-t Coords)
Example Problem (Polar Coords)
Section B: Work and Energy Section 3.6 – Work and Kinetic Energy We have used the Force, Mass & Acceleration approach to determine the motion of a particle via the Equation of Motion. This gives the relationship between instantaneous acceleration of the particle and the unbalanced force. We determine the motion (velocity and position) by integrating the acceleration using the appropriate kinematic equations.
There are 2 classes of problems in which the cumulative effects of the unbalanced forces that act on the particle are of interest to us: Cases involving the integration of the forces with respect to the displacement of the particle, and Cases involving the integration of the forces with respect to the time they are applied. We can incorporate the results of these integrations directly into the governing equations of motion so that it becomes unnecessary to solve directly for the acceleration.
Integration with respect to the displacement leads to the equations of Work and Energy. Integration with respect to time leads to the equations of Impulse and Momentum.
Definition of Work A force F acts on a particle at A which moves along the path shown to A. The position vector r is measured from an arbitrary, convenient reference point O. dr is the differential displacement associated with an infinitesimal movement from A to A.
Definition of Work The work dU done by F during the displacement dr is defined as: Where is the angle between F and dr, and ds = |dr| Note: work is a scalar (dot product) quantity.
Definition of Work The work dU done by F during the displacement dr is defined as: This may be interpreted as the displacement ds multiplied by the component of force in the direction of the displacement Ft = F cos, or This may be interpreted as the force F multiplied by the component of the displacement in the direction of the force ds cos. It should be noted that Fn = F sin (the component of the force normal to the displacement) does no work.
Sign Convention Work done on a particle is +ive if the working component of the force Ft is in the direction of the displacement. Work done on a particle is –ive if the working component of the force Ft is in the direction opposite the displacement.
Consider a particle of mass m falling under the influence of gravity, and subject to an aerodynamic drag proportional to v2. This is a rectilinear problem. does –ive work does +ive work 2nd order ODE
Units of Work The SI unit of work is the Joule, J. 1 J = 1 Nm We should always use J for work and energy units, the notation Nm will be used for moments which as we will see are a vector quantity.
Calculation of Work During a finite movement of the particle the force does an amount of work given by:
Calculation of Work In order to carry out this integral we must know the relationship between the force components and their respective component displacements, or the relationship between Ft and s. For an analytical solution an analytical expression of Ft(s) is required. This is not always available.
The work done during a finite movement from P1 to P2 is: Let us substitute Newton’s 2nd Law:
Principle of Work and Kinetic Energy