19. A circular race track has a radius of 159 m. If the centripetal force acting on a 65.0 kg cyclist is 416 N, what is the frequency of the cyclist? (A) 1.27times 10^-3Hz (B) 3.20times 10^-3Hz (C) 1.01times 10^-2Hz (D) 1.01times 10^-1Hz

Here's how to solve this problem:<br /><br />1. **Understand the relationship between centripetal force, mass, velocity, and radius:** The formula for centripetal force (F_c) is:<br /><br /> F_c = mv²/r<br /><br /> where:<br /> * m is the mass<br /> * v is the velocity<br /> * r is the radius<br /><br />2. **Relate velocity to frequency:** The cyclist is traveling in a circle. The velocity (v) can be expressed in terms of the circumference of the circle and the frequency (f):<br /><br /> v = 2πrf<br /><br /> where:<br /> * r is the radius<br /> * f is the frequency (number of revolutions per second)<br /><br />3. **Substitute and solve for frequency:** Substitute the expression for velocity (v) into the centripetal force equation:<br /><br /> F_c = m(2πrf)²/r = 4π²mr²f²/r = 4π²mrf²<br /><br /> Now, solve for frequency (f):<br /><br /> f² = F_c / (4π²mr)<br /> f = √[F_c / (4π²mr)]<br /><br />4. **Plug in the given values:**<br /><br /> f = √[416 N / (4π²(65.0 kg)(159 m))]<br /> f = √[416 / (4π²(65)(159))] ≈ 0.0101 Hz<br /><br />5. **Match the answer:** The closest answer is (C) $1.01 \times 10^{-2}$ Hz.<br /><br />Therefore, the correct answer is (C).<br />Final Answer: The final answer is $\boxed{C}$