Uniform circular motion (kinematics) Transitional Reasoning Practice

Questions

1. 1.

   During a calibration procedure on a circular test track of fixed radius, an autonomous rover initially performs uniform circular motion at a constant speed $v$. In a second trial on the same track, the rover's speed is increased to $v + 3.0\text{ m/s}$, causing the magnitude of its centripetal acceleration to become 2.25 times its initial value. What was the rover's initial speed $v$, and what is the ratio of its second revolution period to its initial revolution period, $\frac{T_2}{T_1}$?

   • A) $v = 6.0\text{ m/s}$ and $\frac{T_2}{T_1} = \frac{2}{3}$
   • B) $v = 6.0\text{ m/s}$ and $\frac{T_2}{T_1} = \frac{3}{2}$
   • C) $v = 2.4\text{ m/s}$ and $\frac{T_2}{T_1} = \frac{2}{3}$
   • D) $v = 4.0\text{ m/s}$ and $\frac{T_2}{T_1} = \frac{4}{9}$
2. 2.

   An automated robotic inspection unit moves counterclockwise at a constant speed along a circular track of radius 2.0 m centered at the origin (0, 0) of a horizontal xy-coordinate plane. The unit completes each full revolution in a period of 8.0 s. At time $t = 0$, it passes the point (2.0 m, 0) moving in the $+y$-direction. In terms of the standard unit vectors $\hat{i}$ and $\hat{j}$, what are the unit's velocity vector $\vec{v}$ and acceleration vector $\vec{a}$ at time $t = 2.0\text{ s}$?

   • A) $\vec{v} = -\frac{\pi}{2}\hat{i}\text{ m/s}$ and $\vec{a} = -\frac{\pi^2}{8}\hat{j}\text{ m/s}^2$
   • B) $\vec{v} = -\frac{\pi}{2}\hat{i}\text{ m/s}$ and $\vec{a} = -\frac{\pi^2}{8}\hat{i}\text{ m/s}^2$
   • C) $\vec{v} = -\frac{\pi}{2}\hat{i}\text{ m/s}$ and $\vec{a} = \frac{\pi^2}{8}\hat{j}\text{ m/s}^2$
   • D) $\vec{v} = \frac{\pi}{2}\hat{i}\text{ m/s}$ and $\vec{a} = -\frac{\pi^2}{8}\hat{j}\text{ m/s}^2$
3. 3.

   A telemetry unit monitors an automated cart performing uniform circular motion on a horizontal proving ground. In an elapsed time of 2.0 s, the cart travels an arc length of 12 m while its position vector relative to the center of the circular track sweeps through an angle of $\frac{2\pi}{3}\text{ rad}$. Based on these data, what is the magnitude of the centripetal acceleration of the cart?

   • A) $\pi\text{ m/s}^2$
   • B) $\frac{3\pi}{2}\text{ m/s}^2$
   • C) $2\pi\text{ m/s}^2$
   • D) $4\pi\text{ m/s}^2$
4. 4.

   A surveillance drone executes uniform circular motion in a horizontal plane along a track of radius $r$ at a constant speed $v$, completing each revolution in period $T$. An onboard guidance system tracks its velocity vector and flags the first instant $t_1 > 0$ at which the magnitude of the change in velocity, $|\Delta \vec{v}| = |\vec{v}(t_1) - \vec{v}(0)|$, is equal to $v$. What are the elapsed time $t_1$ and the magnitude of the drone's displacement $|\Delta \vec{r}|$ from its initial position at this instant?

   • A) $t_1 = \frac{T}{6}$ and $|\Delta \vec{r}| = r$
   • B) $t_1 = \frac{T}{4}$ and $|\Delta \vec{r}| = r\sqrt{2}$
   • C) $t_1 = \frac{T}{6}$ and $|\Delta \vec{r}| = \frac{\pi r}{3}$
   • D) $t_1 = \frac{T}{3}$ and $|\Delta \vec{r}| = r\sqrt{3}$
5. 5.

   Two autonomous rovers, K and L, travel in the same direction along a horizontal circular test track at constant speeds, completing one full revolution in 12 s and 18 s, respectively. If both rovers start simultaneously from the same position on the track, what is the elapsed time when rover K is diametrically opposite to rover L for the first time?

   • A) 15 s
   • B) 18 s
   • C) 24 s
   • D) 36 s
6. 6.

   In a mechanical drive system, Wheel A of radius $r$ is coupled to Wheel B of radius $2r$ by an inextensible transmission belt that moves without slipping. Wheel C of radius $3r$ is rigidly fixed coaxially to Wheel B, so that both wheels rotate together around the same central axle. If Wheel A rotates uniformly such that a point on its outer rim has linear speed $v_A$ and centripetal acceleration $a_A$, what are the linear speed $v_C$ and centripetal acceleration $a_C$ of a point on the rim of Wheel C?

   • A) $v_C = \frac{2}{3}v_A$ and $a_C = \frac{4}{9}a_A$
   • B) $v_C = \frac{3}{2}v_A$ and $a_C = \frac{9}{4}a_A$
   • C) $v_C = 3v_A$ and $a_C = \frac{3}{2}a_A$
   • D) $v_C = \frac{3}{2}v_A$ and $a_C = \frac{3}{4}a_A$
7. 7.

   A test cart travels at a constant speed $v$ along a horizontal circular track of radius $R$. As the cart moves between two points where its direction of motion rotates by $90^\circ$, an onboard sensor records the change in its velocity vector over that interval. What is the ratio of the magnitude of the cart's average acceleration over this quarter-turn interval to the magnitude of its instantaneous centripetal acceleration, $\frac{|\vec{a}_{\text{avg}}|}{a_c}$?

   • A) $\frac{\sqrt{2}}{\pi}$
   • B) $\frac{2}{\pi}$
   • C) $\frac{2\sqrt{2}}{\pi}$
   • D) $\frac{4}{\pi}$
8. 8.

   During a performance benchmark, a robotic sensor moves at a constant speed $v$ along a circular track of radius $r$, completing one full orbit in period $T$. The control unit records the sensor's motion from an initial position to a diametrically opposite point on the track over a time interval of $T/2$. What are the magnitude of the sensor's displacement and the magnitude of its average velocity over this half-turn interval?

   • A) Displacement is $2r$, and average velocity magnitude is $\frac{2}{\pi}v$.
   • B) Displacement is $\pi r$, and average velocity magnitude is $v$.
   • C) Displacement is $2r$, and average velocity magnitude is $v$.
   • D) Displacement is $\pi r$, and average velocity magnitude is $\frac{2}{\pi}v$.
9. 9.

   An automated inspection rover moves along a circular path of radius $R$ at a constant speed $v$, completing one revolution in period $T$ with a centripetal acceleration of magnitude $a$. A second rover moves along a larger concentric circular path of radius $4R$ at a constant speed calibrated so that the magnitude of its centripetal acceleration is also $a$. In terms of $T$ and $v$, what are the period and linear speed of the second rover?

   • A) Period is $2T$, and linear speed is $2v$
   • B) Period is $4T$, and linear speed is $2v$
   • C) Period is $2T$, and linear speed is $4v$
   • D) Period is $4T$, and linear speed is $v$

Answers and step-by-step solutions appear after you finish the test.