Question

In the Thunder Sphere, a motorcycle moves on the inside of a sphere, traveling in a horizontal circle along the equator of the sphere. The motorcycle maintains a speed of \(15.11 \mathrm{~m} / \mathrm{s}\), and the coefficient of static friction between the tires of the motorcycle and the inner surface of the sphere is 0.4741 . What is the maximum radius that the sphere can have if the motorcycle is not to fall?

Short answer

Answer: The maximum radius of the sphere is 49.11 meters.

Step-by-step solution

Identify the Forces Acting on the Motorcycle

First, we need to identify the forces acting on the motorcycle. There are three forces of interest: 1. Centripetal force (Fc): The force required to maintain circular motion along the horizontal circle. 2. Gravitational force (Fg): The force due to gravity acting on the motorcycle, pulling it towards the center of the Earth vertically. 3. Static friction force (Fs): The force of friction between the tires of the motorcycle and the inner surface of the sphere.

Calculate the Centripetal Force

The centripetal force (Fc) is given by the formula: Fc = (m * v^2) / r where m is the mass of the motorcycle, v is the speed (given as 15.11 m/s), and r is the radius of the sphere. Since we are not given the mass of the motorcycle, we can assume that m = 1 without loss of generality. Fc = (1 * 15.11^2) / r Fc = 228.33 / r

Calculate the Gravitational Force

The gravitational force (Fg) acting on the motorcycle is given by the formula: Fg = m * g where m is the mass of the motorcycle and g is the acceleration due to gravity (approximately 9.81 m/s^2). Again, we assume m = 1. Fg = 1 * 9.81 Fg = 9.81 N

Calculate the Maximum Static Friction Force

The maximum static friction force (Fs) between the tires of the motorcycle and the inner surface of the sphere is given by the formula: Fs = μs * Fg where μs is the coefficient of static friction (given as 0.4741) and Fg is the gravitational force. Fs = 0.4741 * 9.81 Fs = 4.65 N

Set up the Inequality for Maximum Sphere Radius

For the motorcycle not to fall, the static friction force should be equal to or exceeding the centripetal force. Therefore, we have: Fs >= Fc By plugging in the values calculated in Step 2 and Step 4, we have the inequality: 4.65 >= 228.33 / r

Solve for the Maximum Radius

By rearranging the inequality and solving for r, we get: r <= (228.33 / 4.65) r <= 49.11 m Thus, the maximum radius that the sphere can have, if the motorcycle is not to fall, is 49.11 meters.

Key concepts

Centripetal Force

Centripetal force plays a critical role in the analysis of any object moving in a circular path, including the scenario of a motorcycle defying gravity in the Thunder Sphere.

This force is responsible for keeping an object in circular motion on a constant path and acts towards the center of the circle. It's what prevents the motorcycle from shooting out tangentially from its circular path within the sphere.

The centripetal force equation, given by \( F_c = \frac{m \cdot v^2}{r} \), where \( m \) is the mass of the object, \( v \) is the velocity, and \( r \) is the radius, is pivotal to understanding the relationship between speed, the radius of the path, and the needed force to maintain the motion.

Now, in our example, since the mass of the motorcycle doesn't change, we focus on the velocity and how a larger radius \( r \) would require a greater centripetal force to maintain the same velocity. This ultimately governs the maximum radius the sphere can have without causing the motorcycle to fall, as it's directly tied to the maximum force friction can provide.

Static Friction

In the spectacle of the Thunder Sphere, static friction is the unsung hero ensuring the motorcycle stays on track.

Static friction is the force that resists the initiation of sliding motion between two surfaces in contact. It is the stronger cousin in the friction family, typically higher than kinetic (sliding) friction, and plays an essential role when the motorcycle tires adhere to the sphere walls.

The maximum value static friction can take is quantified by the equation \( F_s = \mu_s \cdot F_g \), with \( \mu_s \) representing the coefficient of static friction, and \( F_g \) the gravitational force. As the coefficient increases, so does the ability of the motorcycle to stick to the sphere's surface without sliding down, essentially allowing larger sphere radii for the same gravitational force.

This highlights a critical point—there's a balancing act between the velocity, the gravitational pull, and the static frictional force that allows the motorcycle to defy gravity and whirl around the sphere without losing grip.

Gravitational Force

Gravitational force is the universal pull that attracts two bodies towards each other, an invisible string that keeps the motorcycle from leaving the Earth—and also the Thunder Sphere.

Described by the formula \( F_g = m \cdot g \), where \( g \), approximately \( 9.81 \, \text{m/s}^2 \), is the acceleration due to gravity, this force provides the weight that the static friction uses to anchor the motorcycle to the sphere's surface. While gravity tries to pull the motorcycle toward the center of the Earth, friction exploits this downward force to ensure that the motorcycle doesn't careen out of the sphere.

A profound understanding of gravitational force is not only key to predicting how objects will interact with our planet but also essential in understanding phenomena like orbiting satellites, planetary motion, and indeed, daring motorcycle stunts within spherical cages.