Circular Motion: Vertical Circles and Banked Turns

Circular Motion

Minimum Coefficient of Friction for a Car on a Flat Curve

When a car travels around a flat, unbanked curve, friction between the tires and the road provides the necessary centripetal force to keep the car moving in a circle. The minimum coefficient of static friction required can be calculated as follows:

• Centripetal Force (Fc): The force required to keep an object moving in a circle of radius R at speed v is $F_c = \frac{mv^2}{R}$.

• Normal Force (FN): On a horizontal surface, $F_N = mg$.

• Frictional Force (fs): The maximum static friction is $f_s = \mu_s F_N$.

• Setting $f_s = F_c$ gives $\mu_s mg = \frac{mv^2}{R}$, so $\mu_s = \frac{v^2}{gR}$.

• Example: For $v = 36.0\ \mathrm{m/s}$, $R = 300\ \mathrm{m}$, $g = 9.8\ \mathrm{m/s^2}$, $\mu_s = 0.441$.

Vertical Circles

Vertical circular motion involves objects moving in a circle in a vertical plane, such as a motorcycle inside a sphere or a pendulum. The forces acting on the object change depending on its position in the circle.

• Apparent Weight: The normal force experienced by the object, which can differ from its actual weight due to acceleration.

• At the Bottom: $F_N = F_c + F_g$; the normal force is greatest, and the object feels heavier.

• At the Sides: $F_N = F_c$; gravity does not contribute to the normal force.

• At the Top: $F_N = F_c - F_g$; if $F_N < 0$, the object loses contact and falls.

• Minimum Speed at the Top: To maintain contact, $F_N \geq 0$, so $F_c \geq F_g$.

Example: Minimum Speed at the Top of a Vertical Circle

• For a circle of radius $R$, $F_c = F_g$ at the minimum speed.

• $\frac{mv_{min}^2}{R} = mg \implies v_{min} = \sqrt{gR}$

• For $R = 6.0\ \mathrm{m}$, $v_{min} = 7.668\ \mathrm{m/s}$

Conservation of Energy in Vertical Circles

As an object moves around a vertical circle, its kinetic and potential energies change, but the total mechanical energy is conserved (if friction is negligible).

• At the Bottom: Maximum kinetic energy, minimum potential energy.

• At the Top: Minimum kinetic energy, maximum potential energy.

• Energy Conservation: $\frac{1}{2}mv_{max}^2 = \frac{1}{2}mv_{min}^2 + mg(2R)$

Banked Turns

Banked turns are designed so that the normal force provides the necessary centripetal force for circular motion, reducing reliance on friction. This is common in highways and racetracks.

• Normal Force Components: The normal force $F_N$ can be resolved into horizontal ($F_{Nx}$) and vertical ($F_{Ny}$) components.

• Vertical Equilibrium: $F_{Ny} = F_g \implies F_N \cos \theta = mg$

• Horizontal (Centripetal) Force: $F_{Nx} = F_c \implies F_N \sin \theta = \frac{mv^2}{r}$

• Bank Angle Formula: $\tan \theta = \frac{v^2}{rg}$

• Maximum Speed (No Friction): $v = \sqrt{rg \tan \theta}$

• Example: For $r = 100\ \mathrm{m}$, $\theta = 30^\circ$, $v_{max} = 23.8\ \mathrm{m/s}$

Banked Turns with Friction

When friction is present, it can increase the maximum speed at which a car can safely navigate a banked turn. The frictional force adds to the normal force components, allowing for greater centripetal force.

• Frictional Force: Acts parallel to the surface, increasing both the x and y components of the net force.

• Designing Safe Turns: Engineers calculate the required banking angle to ensure safety even in low-friction conditions (e.g., icy roads).

• Example: For $r = 75.0\ \mathrm{m}$, $v = 24.59\ \mathrm{m/s}$, the required bank angle is $39.4^\circ$.

Summary Table: Forces in Circular Motion

Position	Normal Force ($F_N$)	Apparent Weight	Condition
Bottom of Vertical Circle	$F_N = F_c + F_g$	Heavier	Maximum $F_N$
Side of Vertical Circle	$F_N = F_c$	Normal	Gravity perpendicular to motion
Top of Vertical Circle	$F_N = F_c - F_g$	Lighter or zero	Minimum $F_N$

Key Equations

• $F_c = \frac{mv^2}{R}$

• $\mu_s = \frac{v^2}{gR}$

• $v_{min} = \sqrt{gR}$

• $\tan \theta = \frac{v^2}{rg}$

• $v = \sqrt{rg \tan \theta}$

Additional info: The image included shows a car navigating a banked turn, which visually reinforces the explanation of how banked turns use the normal force to provide centripetal force, reducing reliance on friction.