An automobile travels counterclockwise around a circular roundabout of radius

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Torque and Rotational Dynamics · Torque and Rotational Dynamics

Problem

45\,\text{m}

. The car’s speed

v(t)

(in

\text{m s}^{-1}

) and the central angle

\phi(t)

(in radians) that locates the car on the roundabout are both differentiable functions of time

t

seconds. At a particular instant, the car is located so that

\phi = \pi/6

, its speed is

18\,\text{m s}^{-1}

, and its tangential acceleration

\frac{dv}{dt}

2.5\,\text{m s}^{-2}

. (a) Determine the instantaneous angular velocity

\frac{d\phi}{dt}

of the car about the center of the roundabout at that instant. Express your answer in

\text{rad s}^{-1}

. (b) Derive a formula for the angular acceleration

\frac{d^2\phi}{dt^2}

in terms of

v

\frac{dv}{dt}

, and the radius of the roundabout. Then compute its value at the given instant, specifying units. (c) Find the magnitude of the car’s total acceleration vector at this instant and classify its two perpendicular components. (d) Suppose safety regulations limit the magnitude of the car’s total acceleration to

6\,\text{m s}^{-2}

. Determine the greatest tangential acceleration

\frac{dv}{dt}

the driver could apply at the given speed and still satisfy the regulation.

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