A scientific satellite of mass $m$ orbits a planet of mass $M$ in a uniform circular orbit of radius $r$ at speed $v$. Ground controllers maneuver the satellite into a new stable circular orbit of radius $4r$. Which statement correctly describes how the gravitational centripetal force and the orbital speed of the satellite in the new orbit compare to their initial values?

A) The centripetal force decreases to $\frac{1}{16}$ of its original value, and the orbital speed decreases to $\frac{1}{2}$ of its original value.

B) The centripetal force decreases to $\frac{1}{4}$ of its original value, and the orbital speed decreases to $\frac{1}{2}$ of its original value.

C) The centripetal force decreases to $\frac{1}{16}$ of its original value, and the orbital speed decreases to $\frac{1}{4}$ of its original value.

D) The centripetal force decreases to $\frac{1}{4}$ of its original value, and the orbital speed remains unchanged.

An aerospace engineer analyzes an airplane of mass $m$ executing a steady horizontal turn of radius $R$ at a constant speed $v$. The airplane banks its wings at an angle $\theta$ to the horizontal so that the aerodynamic lift force $F_L$, directed perpendicular to the wings, supports the airplane against gravity while providing the centripetal acceleration. Which component of the aerodynamic lift force acts as the centripetal force directed toward the center of the turn, and what is the resulting radius $R$ of the circle?

A) $F_L \cos\theta$ acts as the centripetal force, and the radius is $R = \frac{v^2 \tan\theta}{g}$.

B) $F_L \cos\theta$ acts as the centripetal force, and the radius is $R = \frac{v^2}{g \sin\theta}$.

C) $F_L \sin\theta$ acts as the centripetal force, and the radius is $R = \frac{v^2}{g \tan\theta}$.

D) $F_L \sin\theta$ acts as the centripetal force, and the radius is $R = \frac{g \tan\theta}{v^2}$.

To simulate Earth gravity during long-duration interplanetary travel, engineers design a large cylindrical habitat of radius $R$ that rotates at a constant rate in deep space. An astronaut of mass $m$ stands along the interior of the outer hull, oriented with their feet on the hull and head pointed toward the central rotation axis. Which force acts as the centripetal force to keep the astronaut moving in this circular path, and what tangential speed $v$ must the outer rim maintain to simulate an acceleration of magnitude $g$?

A) The habitat's gravitational attraction provides the centripetal force, and the required speed is $v = \sqrt{g/R}$.

B) A fictitious centrifugal force directed inward provides the centripetal force, and the required speed is $v = gR$.

C) The normal force exerted inward by the hull floor provides the centripetal force, and the required speed is $v = \sqrt{gR}$.

D) The static friction force parallel to the hull floor provides the centripetal force, and the required speed is $v = \sqrt{2gR}$.

In a laboratory experiment, a puck of mass $m_1$ moves on a frictionless horizontal tabletop in a circle of radius $R$ at a constant speed $v$. A light string attached to the puck passes through a frictionless hole at the center of the table and supports a hanging block of mass $m_2$, which remains stationary. What is the magnitude of the centripetal force $F_c$ acting on the puck, and what is the correct expression for its speed squared, $v^2$?

A) $F_c = m_1 g$ and $v^2 = \frac{m_1}{m_2} g R$

B) $F_c = m_2 g$ and $v^2 = \frac{m_2}{m_1} g R$

C) $F_c = m_2 g$ and $v^2 = \frac{m_1}{m_2} g R$

D) $F_c = (m_1 + m_2)g$ and $v^2 = \frac{m_2}{m_1 + m_2} g R$

An amusement park ride consists of a vertical hollow cylinder of radius $R$ rotating at a constant angular speed about its central axis. A rider of mass $m$ stands against the inner wall, and once the cylinder reaches a tangential speed $v$, the floor drops away beneath the rider. If the rider remains held against the wall without sliding downward, which force provides the centripetal acceleration, and what is the minimum coefficient of static friction $\mu_s$ between the rider and the wall to prevent slipping?

A) The static friction force directed vertically upward; $\mu_s = \frac{v^2}{g R}$

B) The static friction force directed vertically upward; $\mu_s = \frac{g R}{v^2}$

C) The normal force directed toward the rotation axis; $\mu_s = \frac{g R}{v^2}$

D) The normal force directed toward the rotation axis; $\mu_s = \frac{v^2}{g R}$

A civil engineer evaluates safety guidelines for vehicles traveling over a humpback bridge modeled as a vertical circular arc of radius $R$. An automobile of mass $m$ crosses the highest point of the bridge at a constant speed $v$ while maintaining contact with the road. Which expression correctly gives the normal force $F_N$ exerted by the road on the automobile at this peak, and what is the maximum speed $v_{\text{max}}$ it can travel without becoming airborne?

A) $F_N = m\left(g + \frac{v^2}{R}\right)$ and $v_{\text{max}} = \sqrt{gR}$

B) $F_N = m\left(g - \frac{v^2}{R}\right)$ and $v_{\text{max}} = \sqrt{gR}$

C) $F_N = m\left(\frac{v^2}{R} - g\right)$ and $v_{\text{max}} = \sqrt{2gR}$

D) $F_N = mg$ and $v_{\text{max}} = \frac{gR}{2}$

A small bob of mass $m$ is attached to a fixed support by a light string and moves in a horizontal circle of radius $r$ at a constant speed $v$. Throughout the motion, the string remains taut and maintains a constant angle $\theta$ with the vertical. Neglecting air resistance, which option correctly identifies the source of the centripetal force and gives the correct expression for $v^2$?

A) The horizontal component of the string tension; $v^2 = rg \tan\theta$

B) The horizontal component of the string tension; $v^2 = rg \sin\theta$

C) The entire tension force acting along the string; $v^2 = rg \cos\theta$

D) The net resultant of gravity and a fictitious outward force; $v^2 = \frac{rg}{\tan\theta}$

A roller-coaster car of mass $m$ travels along a track and passes through the lowest point of a vertical circular dip of radius $R$ at speed $v$. A force sensor beneath the wheels records the upward normal force exerted by the track on the car at this exact point. Which expression correctly gives the magnitude of the normal force, and what explains its relationship to the car's weight $mg$?

A) $mg + \frac{m v^2}{R}$, because the normal force must counteract gravity and supply the upward centripetal acceleration.

B) $\frac{m v^2}{R} - mg$, because gravity acts in the same direction as the inward centripetal acceleration.

C) $\frac{m v^2}{R}$, because the centripetal force is a separate contact force that replaces the normal force at the dip.

D) $\sqrt{(mg)^2 + \left(\frac{m v^2}{R}\right)^2}$, because the gravitational force and centripetal force act perpendicularly to each other.

A student investigates circular motion by placing two identical small blocks, $P$ and $Q$, on a rough horizontal turntable that rotates at a constant angular speed. Block $P$ is placed at a distance $r$ from the central axis, while block $Q$ is placed at a distance $2r$, and neither block slips relative to the surface. How does the magnitude of the net horizontal force acting on block $Q$ compare to that acting on block $P$, and which force provides it?

A) It is twice as large for block $Q$, provided by static friction directed toward the rotation axis.

B) It is four times as large for block $Q$, provided by static friction directed toward the rotation axis.

C) It is half as large for block $Q$, provided by kinetic friction directed opposite to the tangential motion.

D) It is equal for both blocks, provided by the normal force directed radially outward from the axis.

Centripetal force Transitional Reasoning Practice

Physics · Forces & Newton's Laws · Physics · Circular Motion & Gravitation

10 Physics practice questions on Centripetal force. Questions that build towards applied reasoning.

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1.
A scientific satellite of mass $m$ orbits a planet of mass $M$ in a uniform circular orbit of radius $r$ at speed $v$ . Ground controllers maneuver the satellite into a new stable circular orbit of radius $4r$ . Which statement correctly describes how the gravitational centripetal force and the orbital speed of the satellite in the new orbit compare to their initial values?
- A) The centripetal force decreases to $\frac{1}{16}$ of its original value, and the orbital speed decreases to $\frac{1}{2}$ of its original value.
- B) The centripetal force decreases to $\frac{1}{4}$ of its original value, and the orbital speed decreases to $\frac{1}{2}$ of its original value.
- C) The centripetal force decreases to $\frac{1}{16}$ of its original value, and the orbital speed decreases to $\frac{1}{4}$ of its original value.
- D) The centripetal force decreases to $\frac{1}{4}$ of its original value, and the orbital speed remains unchanged.
2.
An aerospace engineer analyzes an airplane of mass $m$ executing a steady horizontal turn of radius $R$ at a constant speed $v$ . The airplane banks its wings at an angle $\theta$ to the horizontal so that the aerodynamic lift force $F_L$ , directed perpendicular to the wings, supports the airplane against gravity while providing the centripetal acceleration. Which component of the aerodynamic lift force acts as the centripetal force directed toward the center of the turn, and what is the resulting radius $R$ of the circle?
- A) $F_L \cos\theta$ acts as the centripetal force, and the radius is $R = \frac{v^2 \tan\theta}{g}$ .
- B) $F_L \cos\theta$ acts as the centripetal force, and the radius is $R = \frac{v^2}{g \sin\theta}$ .
- C) $F_L \sin\theta$ acts as the centripetal force, and the radius is $R = \frac{v^2}{g \tan\theta}$ .
- D) $F_L \sin\theta$ acts as the centripetal force, and the radius is $R = \frac{g \tan\theta}{v^2}$ .
3.
To simulate Earth gravity during long-duration interplanetary travel, engineers design a large cylindrical habitat of radius $R$ that rotates at a constant rate in deep space. An astronaut of mass $m$ stands along the interior of the outer hull, oriented with their feet on the hull and head pointed toward the central rotation axis. Which force acts as the centripetal force to keep the astronaut moving in this circular path, and what tangential speed $v$ must the outer rim maintain to simulate an acceleration of magnitude $g$ ?
- A) The habitat's gravitational attraction provides the centripetal force, and the required speed is $v = \sqrt{g/R}$ .
- B) A fictitious centrifugal force directed inward provides the centripetal force, and the required speed is $v = gR$ .
- C) The normal force exerted inward by the hull floor provides the centripetal force, and the required speed is $v = \sqrt{gR}$ .
- D) The static friction force parallel to the hull floor provides the centripetal force, and the required speed is $v = \sqrt{2gR}$ .
4.
In a laboratory experiment, a puck of mass $m_1$ moves on a frictionless horizontal tabletop in a circle of radius $R$ at a constant speed $v$ . A light string attached to the puck passes through a frictionless hole at the center of the table and supports a hanging block of mass $m_2$ , which remains stationary. What is the magnitude of the centripetal force $F_c$ acting on the puck, and what is the correct expression for its speed squared, $v^2$ ?
- A) $F_c = m_1 g$ and $v^2 = \frac{m_1}{m_2} g R$
- B) $F_c = m_2 g$ and $v^2 = \frac{m_2}{m_1} g R$
- C) $F_c = m_2 g$ and $v^2 = \frac{m_1}{m_2} g R$
- D) $F_c = (m_1 + m_2)g$ and $v^2 = \frac{m_2}{m_1 + m_2} g R$
5.
An amusement park ride consists of a vertical hollow cylinder of radius $R$ rotating at a constant angular speed about its central axis. A rider of mass $m$ stands against the inner wall, and once the cylinder reaches a tangential speed $v$ , the floor drops away beneath the rider. If the rider remains held against the wall without sliding downward, which force provides the centripetal acceleration, and what is the minimum coefficient of static friction $\mu_s$ between the rider and the wall to prevent slipping?
- A) The static friction force directed vertically upward; $\mu_s = \frac{v^2}{g R}$
- B) The static friction force directed vertically upward; $\mu_s = \frac{g R}{v^2}$
- C) The normal force directed toward the rotation axis; $\mu_s = \frac{g R}{v^2}$
- D) The normal force directed toward the rotation axis; $\mu_s = \frac{v^2}{g R}$
6.
A civil engineer evaluates safety guidelines for vehicles traveling over a humpback bridge modeled as a vertical circular arc of radius $R$ . An automobile of mass $m$ crosses the highest point of the bridge at a constant speed $v$ while maintaining contact with the road. Which expression correctly gives the normal force $F_N$ exerted by the road on the automobile at this peak, and what is the maximum speed $v_{\text{max}}$ it can travel without becoming airborne?
- A) $F_N = m\left(g + \frac{v^2}{R}\right)$ and $v_{\text{max}} = \sqrt{gR}$
- B) $F_N = m\left(g - \frac{v^2}{R}\right)$ and $v_{\text{max}} = \sqrt{gR}$
- C) $F_N = m\left(\frac{v^2}{R} - g\right)$ and $v_{\text{max}} = \sqrt{2gR}$
- D) $F_N = mg$ and $v_{\text{max}} = \frac{gR}{2}$
7.
A small bob of mass $m$ is attached to a fixed support by a light string and moves in a horizontal circle of radius $r$ at a constant speed $v$ . Throughout the motion, the string remains taut and maintains a constant angle $\theta$ with the vertical. Neglecting air resistance, which option correctly identifies the source of the centripetal force and gives the correct expression for $v^2$ ?
- A) The horizontal component of the string tension; $v^2 = rg \tan\theta$
- B) The horizontal component of the string tension; $v^2 = rg \sin\theta$
- C) The entire tension force acting along the string; $v^2 = rg \cos\theta$
- D) The net resultant of gravity and a fictitious outward force; $v^2 = \frac{rg}{\tan\theta}$
8.
A roller-coaster car of mass $m$ travels along a track and passes through the lowest point of a vertical circular dip of radius $R$ at speed $v$ . A force sensor beneath the wheels records the upward normal force exerted by the track on the car at this exact point. Which expression correctly gives the magnitude of the normal force, and what explains its relationship to the car's weight $mg$ ?
- A) $mg + \frac{m v^2}{R}$ , because the normal force must counteract gravity and supply the upward centripetal acceleration.
- B) $\frac{m v^2}{R} - mg$ , because gravity acts in the same direction as the inward centripetal acceleration.
- C) $\frac{m v^2}{R}$ , because the centripetal force is a separate contact force that replaces the normal force at the dip.
- D) $\sqrt{(mg)^2 + \left(\frac{m v^2}{R}\right)^2}$ , because the gravitational force and centripetal force act perpendicularly to each other.
9.
A student investigates circular motion by placing two identical small blocks, $P$ and $Q$ , on a rough horizontal turntable that rotates at a constant angular speed. Block $P$ is placed at a distance $r$ from the central axis, while block $Q$ is placed at a distance $2r$ , and neither block slips relative to the surface. How does the magnitude of the net horizontal force acting on block $Q$ compare to that acting on block $P$ , and which force provides it?
- A) It is twice as large for block $Q$ , provided by static friction directed toward the rotation axis.
- B) It is four times as large for block $Q$ , provided by static friction directed toward the rotation axis.
- C) It is half as large for block $Q$ , provided by kinetic friction directed opposite to the tangential motion.
- D) It is equal for both blocks, provided by the normal force directed radially outward from the axis.
10.
A test engineer investigates the lateral friction force acting on a vehicle of mass $m$ moving at a constant speed $v$ along a flat, horizontal circular track of radius $r$ . Under these baseline conditions, the vehicle completes the turn without skidding under a required horizontal centripetal force of magnitude $F$ . If a second vehicle of mass $2m$ travels at a speed of $2v$ along a track of radius $2r$ , what is the magnitude of the horizontal net force required to keep it on its circular path?
- A) $F$
- B) $2F$
- C) $4F$
- D) $8F$

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