(a) Show that the position of a particle on a circle of radius R with its center at the origin is r → = R (cos θî + sin θĵ ), where θ is the angle the position vector makes with the x -axis. (b) If the particle moves with constant speed v starting on the x -axis at t = 0, find an expression for θ in terms of time t and the period T to complete a full circle, (c) Differentiate the position vector twice with respect to time to find the acceleration, and show that its magnitude is given by Equation 3.16 and its direction is toward the center of the circle.
(a) Show that the position of a particle on a circle of radius R with its center at the origin is r → = R (cos θî + sin θĵ ), where θ is the angle the position vector makes with the x -axis. (b) If the particle moves with constant speed v starting on the x -axis at t = 0, find an expression for θ in terms of time t and the period T to complete a full circle, (c) Differentiate the position vector twice with respect to time to find the acceleration, and show that its magnitude is given by Equation 3.16 and its direction is toward the center of the circle.
(a) Show that the position of a particle on a circle of radius R with its center at the origin is
r
→
= R(cos θî + sin θĵ), where θ is the angle the position vector makes with the x-axis. (b) If the particle moves with constant speed v starting on the x-axis at t = 0, find an expression for θ in terms of time t and the period T to complete a full circle, (c) Differentiate the position vector twice with respect to time to find the acceleration, and show that its magnitude is given by Equation 3.16 and its direction is toward the center of the circle.
A particle is launched from the origin in R^3 with initial velocity v(0) = < 0, 8, 10 > and undergoes constant acceleration a = < 2, - 1, - 5 > due to the combined forces of gravity and wind.
Where and at what time does the particle hit the "ground" (e.g. the plane z = 0)?
A particle is in uniform circular motion about the origin of an xy coordinate system, moving clockwise with a period of 8.40 s. At one instant, its position vector (from the origin) is r→=(3.70m)î-(4.50m)ĵ. At that instant, what is its velocity in unit-vector notation?
An ice skater is executing a figure-eight, consisting of two identically shaped, tangent circular paths. Throughout the first loop, she increases her speed uniformly, and during the second loop she moves at a constant speed. When the ice skater travels during the first loop, the acceleration vector is not directed towards the center, whereas when the ice skater travels in the second loop, the acceleration vector is directed toward the center.
Is this true or false?
Chapter 3 Solutions
Essential University Physics -Modified MasteringPhysics Access
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