Class Examples

Mapundi Kondwani Banda

MKBanda@sun.ac.zaApplied Mathematics Division - Mathematical Sciences

Stellenbosch University

August 20, 2013

Chapter 12:

Problem 12.164A radar gun at O rotates with the angular velocityof θ̇ = 0.1 rad/s and angular acceleration ofθ̈ = 0.025 rad/s2, at the instant θ = 45◦, as itfollows the motion of the car traveling along thecircular road having a radius of r = 200 m.Determine the magnitudes of velocity andacceleration of the car at this instant.

Problem 12.166

If a particle’s position is described by the polarcoordinates r = (2 sin 2θ) m and θ = (4t) rad,where t is in seconds, determine the radial andtransverse components of its velocity andacceleration when t = 1 s.

Problem 12.168The peg moves in the curved slot defined by thelemniscate, and through the slot in the arm. Atθ = 30◦, the angular velocity is θ̇ = 2 rad/s, and theangular acceleration is θ̈ = 1.5 rad/s2. Determinethe magnitudes of the velocity and acceleration ofpeg P at this instant.

Problem 12.190The box slides down the helical ramp such thatr = 0.5m, θ = (0.5t3) rad, and z = (2 − 0.2t2 m,where t is in seconds. Determine the magnitudes ofthe velocity and acceleration of the box at theinstant θ = 2π rad.

Problem 12.184The slotted arm AB drives pin C through the spiralgroove described by the equation r = (1.5θ) m,where θ is in radians. If the arm starts from restwhen θ = 60◦ and is driven at an angular velocity ofθ̇(4t) rad/s where t is in seconds, determine theradial and transverse components of velocity andacceleration of the pin C when t = 1 s.

Problem 12.160

A particle travels around a limaon, defined by theequation r = b − a cos θ, where a and b areconstants. Determine the particle’s radial andtransverse components of velocity and accelerationas a function of θ and its time derivatives.

Problem 12.202Determine the time needed for the load at B toattain a speed of 8 m/s starting from rest, if thecable is drawn into the motor with an accelerationof 0.2 m/s2.

Problem 12.200The motor at C pulls in the cable with anacceleration aC = (3t2) m/s2 , where t is inseconds.The motor at D draws in its cable ataD = 5 m/s2. If both motors start at the sameinstant from rest when d = 3 m, determine (a) thetime needed for d = 0, and (b) the relative velocityof block A with respect to block B when this occurs.

Problem 12.205If the rope is drawn toward the motor M at a speedof vM = (5t3/2) m/s, where t is in seconds,determine the speed of cylinder A when t = 1 s.

Problem 12.214The man pulls the boy up to the tree limb C bywalking backward. If he starts from rest whenxA = 0 and moves backward with a constantacceleration aA = 0.2 m/s2, determine the speed ofthe boy at the instant yB = 4 m. Neglect the size ofthe limb.When xA = 0, yB = 8 m, so that A and Bare coincident, i.e., the rope is 16 m long.

Problem 12.218The man can row the boat in still water with aspeed of 5m/s. If the river is flowing at 2 m/s,determine the speed of the boat and the angle θ hemust direct the boat so that it travels from A to B .

Problem 12.226

A car is traveling north along a straight road at 50km/h. An instrument in the car indicates that thewind is coming from the east. If the car’s speed is80 km/h, the instrument indicates that the wind iscoming from the north-east. Determine the speedand direction of the wind.

Problem 12.228At the instant shown, the bicyclist at A is travelingat 7 m/s around the curve on the race track whileincreasing his speed at 0.5 m/s2. The bicyclist at Bis traveling at 8.5 m/s along the straight-a-way andincreasing his speed at 0.7 m/s2. Determine therelative velocity and relative acceleration of A withrespect to B at this instant.

Problem 12.229At the instant shown car A is traveling with avelocity of 30 m/s and has an acceleration of 2m/s2 along the highway. At the same instant B istraveling on the trumpet interchange curve with aspeed of 15 m/s, which is decreasing at 0.8 m/s2.Determine the relative velocity and relativeacceleration of B with respect to A at this instant.

Problem 12.235The ship travels at a constant speed of vs = 20 m/sand the wind is blowing at a speed of vw = 10 m/s,as shown. Determine the magnitude and directionof the horizontal component of velocity of thesmoke coming from the smoke stack as it appearsto a passenger on the ship.

Problem 12.236Car A travels along a straight road at a speed of 25m/s while accelerating at 1.5 m/s2. At this sameinstant car C is traveling along the straight roadwith a speed of 30 m/s while decelerating at 3m/s2. Determine the velocity and acceleration ofcar A relative to car C .