Circular Motion in a Vertical Loop (College Board AP® Physics 1: Algebra-Based): Revision Note

Circular motion in a vertical loop

• An object such as a ball on a string is an example of circular motion in a vertical loop

• The forces acting on the ball are:

  • tension in the string

  • gravitational force acting straight downwards

• As the ball moves around the circle:

  • the direction of the tension will change continuously

  • the magnitude of the tension will change continuously, reaching:

    • a maximum value at the bottom

    • a minimum value at the top

• The direction of the gravitational force on the ball never changes

  • Therefore, the resultant centripetal force on the ball changes depending on its position as it rotates around the circle

Forces in a vertical circle

Minimum speed

• At the top of a vertical, circular loop, an object requires a minimum speed to maintain circular motion

• The tension in the string is zero at the top of the circular loop

• At the top of the loop at this minimum speed, the gravitational force is the only force that causes the centripetal acceleration

  • So, $a=g$

Forces at the top of a vertical circle

Derived equation

• The minimum speed required to maintain circular motion at the top of the circular loop is given by:

$v=\sqrt{gr}$

• Where:

  • $v=$ minimum speed at the top of the vertical circular loop, measured in $\mathrm{m}/\mathrm{s}$

  • $g=$acceleration due to gravity at Earth's surface, measured in $\mathrm{m}/{\mathrm{s}}^2$

  • $r=$radius of circle, measured in $\mathrm{m}$

Derivation:

Step 1: Identify the fundamental principles

• Newton's second law

${\overrightarrow{F}}_{net}=m\overrightarrow{a}$

• Where:

  • ${\overrightarrow{F}}_{net}$ = net centripetal force exerted on the object

  • $m$ = mass of the object

  • $\overrightarrow{a}$ = centripetal acceleration of the object

• The net force in this case is the gravitational force or the weight force

• The acceleration in this case is the acceleration due to gravity at the Earth's surface

• Therefore:

$\mathrm{Weight}={\overrightarrow{F}}_g=mg$

• Where:

  • ${\overrightarrow{F}}_g$ = gravitational force exerted on the object

  • $m$ = mass of the object

  • $g$ = acceleration due to gravity

• The magnitude of the centripetal acceleration of an object moving in a circular path is given by:

$a_c=\frac{v^2}{r}$

• Where:

  • $a_c$ = magnitude of the centripetal acceleration

  • $v$ = tangential or linear speed

  • $r$ = radius of circular path

Step 2: Apply the specific conditions

• When an object is moving in a circular loop, the net centripetal force is calculated using the tangential speed and the radius of the circular loop

  • Substitute the centripetal acceleration into Newton's second law

${\overrightarrow{F}}_{net}=m\left(\frac{v^2}{r}\right)$

• At the top of a vertical circular loop, the gravitational force is the only force that causes the centripetal acceleration

  • The tension in the string is zero

${\overrightarrow{F}}_g={\overrightarrow{F}}_{net}$

$mg=m\left(\frac{v^2}{r}\right)$

Step 3: Rearrange to obtain an equation for the minimum speed at the top of a vertical circular loop

$\cancel{m}g=\cancel{m}\left(\frac{v^2}{r}\right)$

$g=\left(\frac{v^2}{r}\right)$

$gr=v^2$

$v=\sqrt{gr}$