Centripetal force Transitional Reasoning Download PDF

Questions

1. 1.

   An orbital mechanics team monitors two scientific satellites, $A$ and $B$, in stable circular orbits around a planet. Satellite $B$ orbits at a radius four times that of Satellite $A$, such that $r_B = 4r_A$. Given that the planet's gravitational force provides the necessary centripetal force for circular motion, which of the following correctly compares the orbital speed $v_B$ and centripetal acceleration $a_B$ of Satellite $B$ to those of Satellite $A$?

   • A) $v_B = \frac{1}{2}v_A$ and $a_B = \frac{1}{16}a_A$
   • B) $v_B = \frac{1}{2}v_A$ and $a_B = \frac{1}{4}a_A$
   • C) $v_B = \frac{1}{4}v_A$ and $a_B = \frac{1}{16}a_A$
   • D) $v_B = 2v_A$ and $a_B = \frac{1}{2}a_A$
2. 2.

   A student investigates uniform circular motion using an air-table apparatus where a puck of mass $m$ glides without friction along a horizontal circle of radius $r$. The puck is connected to a light cord that passes through a small frictionless hole at the center of the table and supports a hanging counterweight of mass $M$. If the counterweight remains completely stationary while the puck circles smoothly, which expression correctly gives the required orbital speed $v$ of the puck?

   • A) $v = \sqrt{\frac{M g r}{m}}$
   • B) $v = \sqrt{\frac{m g r}{M}}$
   • C) $v = \sqrt{\frac{(M + m) g r}{m}}$
   • D) $v = \sqrt{\frac{M g}{m r}}$
3. 3.

   A civil engineering team designs an exit ramp as a circular curve of radius $R$ banked at an angle $\theta$ relative to the horizontal. On an icy day with negligible tire-road friction, a car of mass $m$ safely negotiates the curve at the rated design speed without sliding up or down the embankment. Which of the following correctly describes the magnitude of the normal force $F_N$ exerted by the road on the car relative to its weight $mg$, along with the correct physical reasoning?

   • A) $F_N = \frac{mg}{\cos\theta}$ (so $F_N > mg$), because the vertical component of the normal force must balance the car's full weight while its horizontal component supplies the centripetal acceleration.
   • B) $F_N = mg \cos\theta$ (so $F_N < mg$), because the normal force acts perpendicular to the incline and only needs to support the perpendicular component of gravity.
   • C) $F_N = mg$, because the centripetal acceleration is purely horizontal and does not affect the magnitude of the normal force required to support the car.
   • D) $F_N = \frac{mg}{\sin\theta}$ (so $F_N > mg$), because the sine component of the normal force balances gravity while the cosine component supplies the centripetal acceleration.
4. 4.

   Two identical small blocks, $X$ and $Y$, rest on a flat, horizontal turntable at distances $r$ and $2r$ from the central rotational axis, respectively. The coefficient of static friction between each block and the turntable surface is identical. As the turntable slowly increases its angular speed from rest, which block will slip first, and why?

   • A) Block $X$ slips first because the required centripetal force is inversely proportional to the radius of circular motion at a given angular speed.
   • B) Block $Y$ slips first because the required centripetal force is directly proportional to the radius at a shared angular speed, causing it to exceed the maximum static friction first.
   • C) Both blocks slip simultaneously because the maximum static friction force depends only on mass and the coefficient of friction, which are identical for both blocks.
   • D) Block $X$ slips first because it experiences a greater tangential acceleration during the rotational spin-up than block $Y$.
5. 5.

   A small robotic cart of mass $m$ moves in a horizontal circle at a constant angular speed $\omega$ on a frictionless tabletop. The cart is tethered to a fixed central peg by a light horizontal spring with spring constant $k$ and an unstretched length $L_0$. If the spring stretches to establish a circular path of radius $R$ under the condition $k > m\omega^2$, which of the following expressions correctly determines $R$?

   • A) $R = \frac{k L_0}{k - m\omega^2}$
   • B) $R = \frac{k L_0}{k + m\omega^2}$
   • C) $R = \frac{m\omega^2 L_0}{k - m\omega^2}$
   • D) $R = \frac{(k - m\omega^2)L_0}{k}$
6. 6.

   An automotive safety team tests a vehicle of mass $m$ as it drives over the crest of a hill modeled as a circular arc of radius $R$. While passing the highest point of the crest at a constant speed, an onboard accelerometer records a downward centripetal acceleration of magnitude $0.36g$. Assuming the acceleration due to gravity is $g$, what is the magnitude of the normal force exerted by the road on the vehicle at this point?

   • A) $0.36mg$
   • B) $1.36mg$
   • C) $0.64mg$
   • D) $0.80mg$
7. 7.

   In an amusement park rotor ride, riders stand against the inner vertical wall of a hollow cylinder of radius $R$ that rotates at a constant angular velocity about its central vertical axis. Once the ride reaches operating speed, the floor beneath the riders drops away, yet each rider remains held firmly against the wall without sliding downward. Which of the following statements correctly identifies the physical origin of the centripetal force and the force that prevents the rider from falling?

   • A) The centripetal force is provided by static friction directed tangent to the wall, while the inward normal force balances gravity.
   • B) The centripetal force is provided by an outward centrifugal force, while an upward normal force balances gravity.
   • C) The centripetal force is provided by the inward normal force exerted by the wall, while upward static friction balances gravity.
   • D) The centripetal force is provided by gravity acting toward the center of rotation, while upward kinetic friction supports the rider.
8. 8.

   In a physics laboratory investigation, a student sets up a conical pendulum by suspending a small bob of mass $m$ from a light cord. When set into motion, the bob travels in a horizontal circle of radius $r$ at a constant speed while the cord sweeps out a cone, maintaining a constant angle $\theta$ relative to the vertical. Which of the following expressions and physical explanations correctly identifies the magnitude of the centripetal force $F_c$ acting on the bob?

   • A) $F_c = mg \cos\theta$, because the component of the bob's weight perpendicular to the cord provides the inward acceleration.
   • B) $F_c = mg \tan\theta$, because the vertical component of the cord tension balances gravity while the horizontal component acts as the inward net force.
   • C) $F_c = mg \sin\theta$, because the component of the gravitational force along the tilt of the cord pulls the bob toward the center of the circle.
   • D) $F_c = \frac{mg}{\sin\theta}$, because the entire tension along the cord must be directed inward to maintain the circular path.
9. 9.

   A roller coaster car of mass $m$ travels along the inside of a vertical circular loop of radius $R$. At the exact top of the loop, the car moves at the minimum threshold speed $v = \sqrt{gR}$ required to complete the circle without losing contact with the track. Which statement correctly identifies the forces acting on the car and their role at this highest point?

   • A) The track exerts an upward normal force that exactly balances the downward gravitational force, keeping the car in dynamic equilibrium.
   • B) An independent inward centripetal force acts downward in addition to gravity and the normal force to maintain circular motion.
   • C) Gravitational force alone provides the required centripetal acceleration, resulting in a normal force of zero exerted by the track.
   • D) The track exerts its maximum downward normal force on the car to counteract outward centrifugal inertia.