Chapter 2: Space Curves
Section 2.9: Applications to Dynamics
When the parameter for a curve Rp is actually the time, t, then the decomposition for R″ becomes
R..=a=v. T+κ v2 N
where a is the acceleration vector for a particle moving along the curve described by Rt. This is a remarkable result. It says that motion (as described on the left side by the the acceleration vector a) is determined by the geometry on the right side (as described by the geometric quantities T, the unit tangent vector; N, the principal normal; κ, the curvature; and v, the length of R′).
This decomposition for the acceleration vector can be used to solve problems in dynamics where, primarily, the geometry of the path of motion is known. When combined with Newton's second law, F=m a, several interesting applications become accessible.
Together, all four tires of a car in uniform motion around a circular track can exert towards the center of the track, a total frictional force of no more than 450 pounds. The radius of the track is 200 feet and the car weighs 3200 pounds. What is the fastest constant speed the car can sustain without skidding?
At a point on a plane curve Rx the curvature is κ=2, T=i, N=−j, and the force on an object weighing 8 pounds is F=i−2 j pounds. Calculate v and v. at this point.
Find the force of friction required to keep a car on a circular track of radius 300 ft if the car weighs 3200 lbs and travels at the constant speed of 200 ft per second.
A satellite is traveling at a constant speed in a circular orbit 400 miles above the surface of the earth where the acceleration of gravity is 30 ft per sec2. If the radius of the earth is taken as 4000 miles, what is the speed (in miles per hour) of the satellite?
A coin weighing d pounds remains on a disk turning at a uniform speed of 45 rpm if the force on it is not more than d/9 pounds. How far (in inches) from the center of the disk can this coin sit without sliding off?
What is the maximum magnitude of the force needed to cause an object weighing 2 pounds to move at the constant speed of 3 ft per second along the parabola y=x2?
An object of mass 3 kg is traveling counterclockwise around the ellipse 4⁢x2+9⁢y2=36. When it reaches the point 0,−2 its acceleration vector is 3 i+5 j. What is its speed?
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