Chapter 14 : Kinematics Of A Particle – Work and Energy

Chapter Outline The work of a Force Principle of Work and Energy Principle of Work and Energy for a System of Particles Power and Efficiency Conservative Forces and Potential Energy Conservation of Energy

The Work of a Force A force F does work on a particle only when the particle undergoes a displacement in the direction of the force. Consider the force acting on the particle If the particle moves along the path s from position r to new position r’, displacement dr = r’ – r

The Work of a Force Magnitude of dr is represented by ds, differential segment along the path If the angle between tails of dr and F is θ, work dU done by F is a scalar quantity dU = F ds cos θ dU = F·dr (dot product)

The Work of a Force If 0° < θ < 90°, the force component and the displacement has the same sense so that the work is positive If 90° < θ < 180°, the force component and the displacement has the opposite sense so that the work is negative dU = 0 if the force is perpendicular to the displacement since cos 90° = 0 or if the force is applied at a fixed point where displacement = 0 Basic unit for work in SI units is Joule (J)

The Work of a Force This unit combines the units for force and displacement 1 joule of work is done when a force of 1 newton moves 1 meter along its line of action 1J = 1N.m Moment of a force has this same combination of units, however, the concepts of moment and work are in no way related A moment is a vector quantity, whereas work is a scalar

The Work of a Force Work of a Variable Force. If the particle undergoes a finite displacement along its path from r1 to r2 or s1 to s2, the work is determined by integration. If F is expressed as a function of position, F = F(s),

The Work of a Force If the working component of the force, F cos θ, is plotted versus s, the integral in this equation can be interpreted as the area under the curve from position s1 to position s2

The Work of a Force Work of a Constant Force Moving Along a Straight Line. If the force Fc has a constant magnitude and acts at a constant angle θ from its straight line path, then the components of Fc in the direction of displacement is Fc cos θ

The work of Fc represents the area of the rectangle The Work of a Force The work done by Fc when the particle is displaced from s1 to s2 is determined or The work of Fc represents the area of the rectangle

The Work of a Force Work of a Weight. Consider a particle which moves up along the path s from s1 to position s2. At an intermediate point, the displacement dr = dxi +dyj + dzk. Since W = -Wj

The Work of a Force Work done is equal to the magnitude of the particle’s weight times its vertical displacement. If W is downward and ∆y is upward, work is negative If the particle is displaced downward (-∆y), the work of the weight is positive.

The Work of a Force Work of a Spring Force. The magnitude of force developed in a linear elastic spring when the spring is displaced a distance s from its unstretched position is Fs = ks. If the spring is elongated or compressed from a position s1 to s2, the W.D on spring by Fs is positive, since force and displacement are in the same direction.

The Work of a Force This equation represents the trapezoidal area under the line Fs=ks

The Work of a Force If a particle is attached to a spring, then the force Fs exerted on the particle is opposite to that exerted on the spring. The force will do negative work on the particle when the particle is moving so as to further elongate (or compress) the spring.

Example 14.1 The 10-kg block rest on a smooth incline. If the spring is originally stretched 0.5 m, determine the total work done by all forces acting on the block when a horizontal force P = 400 N pushes the block up the plane s = 2 m.

Example 14.1 Horizontal Force P. Since this force is constant, the work is determined using Spring Force Fs. The spring is stretched to its final position s2 = 0.5 + 2 = 2.5 m. The work is negative since force and displacement are in opposite directions.

Example 14.1 The work of Fs is thus Weight W. Weight acts in the opposite direction to its vertical displacement, the work is negative.

Example 14.1 Normal Force NB. This force does no work since it is always perpendicular to the displacement. Total Work. The work of all the forces when the block is displaced 2 m is thus

Principle of Work and Energy For principle of work and energy for the particle, Term on the LHS is the sum of work done by all the forces acting on the particle as the particle moves from point 1 to point 2 Term on the RHS defines the particle’s final and initial kinetic energy Both terms are always positive scalars

Principle of Work and Energy The particle’s initial kinetic energy plus the work done by all the forces acting on the particle as it moves from initial to its final position is equal to the particle’s final kinetic energy For example, if a particle’s initial speed is known and the work of all the forces acting on the particle can be determined, the above eqn provides a direct means of obtaining the final speed v2 of the particle after it undergoes a specified displacement

Principle of Work and Energy If instead v2 is determined by means of the equation of motion, a two step process is necessary, apply ∑Ft = mat to obtain at , then integrate at = v dv/ds to obtain v2. Principle of work and energy cannot be used to determine forces directed normal to the path of the motion since these forces do no work on the particle For curved paths, however, the magnitude of the normal force is s function of speed

Example 14.2 The 17.5-kN automobile is traveling down the 10° inclined road at a speed of 6 m/s. if the driver jams on the brakes, causing his wheels to lock, determine how far s his tires skid on the road. The coefficient of the kinetic friction between the wheels and the road is μk = 0.5

Example 14.2 Work (Free-Body Diagram). The normal force NA does no work since it never undergoes displacement along its line of action. The weight 17.5-kN, is displaced s sin 10° and does positive work. The frictional force FA does both external and internal work. This work is negative since it is in the opposite direction to displacement.

Example 14.2 Applying equation of equilibrium normal to the road, + Principle of Work and Energy.

Example 14.2 Solving for s yields s = 5.75 m

Power and Efficiency Power It is defined as the amount of work performed per unit of time. The power generated by a machine or engine that performs an amount of work dU within a time interval dt is

Power and Efficiency Provided the work dU is expressed by dU = F.dr, then it also possible to write P = F.v Power is a scalar, where in the formulation v represents the velocity of the point which is acted upon by the force F. SI unit for power is watt (W). It is defined as 1 W = 1 J/s = 1 N.m/s

Power and Efficiency Efficiency It is defined as the ratio of the output of useful power produced by the machine to the input of power supplied to the machine

Power and Efficiency If energy applied to the machine occurs during the same time interval at which it is removed, then the efficiency may also be expressed in terms of the ratio of output energy to input energy The efficiency of a machine is always less than 1

Example 14.7 The motor M of the hoist operates with an efficiency of ε = 0.85. Determine the power that must be supplied to the motor to lift the 375-N crate C at the instant point P. The cable has an acceleration of 1.2m/s2, and a velocity of 0.6 m/s

Example 14.7 First we determine the tension in the cable. From the FBD, Since , taking time derivative of this equation, and substituting aP = +1.2 m/s2

Example 14.7 The power output required to draw the cable in at a rate of 0.6 m/s is This power output requires that the motor provide a power input of

Conservative Forces and Potential Energy It is defined by the work done in moving a particle from one point to another that is independent of the path followed by the particle. Two examples are weight of the particle and elastic force of the spring.

Conservative Forces and Potential Energy It is the measure of the amount of work a conservative force will do when it moves from a given position to the datum. Gravitational Potential Energy. If a particle is located a distance y above an arbitrary selected datum, the particle’s weight W has positive gravitational potential energy Vg.

Conservative Forces and Potential Energy W has the capacity of doing positive work when the particle is moved back down to the datum. The particle is located a distance y below the datum, Vg is negative since the weight does negative work when the particle is moved back up to the datum.

Conservative Forces and Potential Energy If y is positive upward, gravitational potential energy of the particle of weight W is Elastic Potential Energy When an elastic spring is elongated or compressed a distance s from its unstretched position, the elastic potential energy Ve can be expressed

Conservative Forces and Potential Energy Ve is always positive since, in the deformed position, the force of the spring has the capacity for always doing positive work on the particle when the spring is returned to its unstretched position.

Conservative Forces and Potential Energy Potential Function. If a particle is subjected to both gravitational and elastic forces, the particle’s potential energy can be expressed as a potential function

Conservative of Energy When a particle is acted upon by a system of both conservative and non-conservative forces, the portion of the work done by the conservative forces can be written in terms of the difference in their potential energies using As a result, the principle of work and energy can be written as

Conservative of Energy (∑U1-2)noncons represents the work of the non-conservative forces acting on the particles. If only conservative forces are applied to the body, this term is zero and we have This equation referred to as the conservation of mechanical energy or simply the conservation of energy

Conservative of Energy It states that during the motion the sum of the particle’s kinetic and potential energies remain constant.

Conservative of Energy System of Particles. If a system of particles is subjected only to conservative forces, then an equation can be written The sum of the particle’s initial kinetic and potential energies is equal to the sum of the particle’s final kinetic and potential energies

Example 14.9 The gantry structure is used to test the response of an airplane during a clash. The plane of mass 8-Mg is hoisted back until θ = 60°, and then pull-back cable AC is released when the plane is at rest. Determine the speed of the plane just before clashing into the ground, θ = 15°. Also, what is the maximum tension developed in the supporting cable during the motion?

Example 14.9 Potential Energy. For convenience, the datum has been established at the top of the gantry.

Example 14.9 Conservation of Energy.

Example 14.9 Equation of Motion. Using data tabulated on the free-body diagram when the plane is at B, +