# Centripetal force

Centripetal force refers to the force that compels an object to traverse a curved trajectory. For instance, centripetal force is exemplified by the Earth’s gravitational pull directed towards its center, which governs the Moon’s orbital path. This force is directed inward, perpendicular to the object’s path of motion. In other words, it is oriented towards the center of the circular path the object is following. This centripetal force is crucial for counteracting the object’s inherent tendency to move in a straight line, allowing it to maintain its trajectory along the curved path.

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## Examples

### Stone

When a stone is rapidly spun around in a sling, it demonstrates the concept of centripetal force, a force directed towards the center of rotation. This force acts as a tether, preventing the stone from breaking free and flying off in a straight line. Instead, the stone continuously changes its direction, compelled to follow the curved path created by the centripetal force. This phenomenon occurs because an object in motion has a natural tendency to move in a straight line, but the centripetal force counters this tendency by constantly redirecting the stone towards the center. In essence, the stone’s circular motion is a delicate balance between the outward inertia seeking to move it straight and the inward centripetal force curving its path, resulting in a graceful and controlled whirl.

### Satellite

A satellite orbiting Earth illustrates the influence of centripetal force. This force counteracts the satellite’s natural forward motion, preventing it from moving in a straight line into space. Instead, the centripetal force continuously redirects the satellite’s trajectory towards Earth, maintaining its circular orbit. The gravitational pull from Earth contributes to this equilibrium, as it provides the centripetal force needed to counterbalance the satellite’s tendency to move straight into space. This harmonious interplay between the satellite’s velocity and the centripetal force ensures a stable and predictable path as it circles our planet.

### Car

When a car navigates a curve on a road, it encounters centripetal force that maintains its path along the curved trajectory. This force acts towards the center of the curve, preventing the car from veering off in a straight line. As the car turns, the tires push against the road surface, generating centripetal force that counters the natural tendency of the car to continue moving straight. The friction between the tires and the road provides the necessary centripetal force, ensuring that the car remains on the intended circular path without skidding or sliding. This balance between the car’s inertia and the centripetal force allows for a controlled and secure maneuver around the curve.

### Planet

The centripetal force plays a vital role in keeping planets, such as Earth, in their orbits around the Sun. This force is provided by the gravitational attraction between the planet and the Sun. As the planet moves along its elliptical path, the Sun’s gravitational pull acts as a centripetal force that constantly pulls the planet towards the center of its orbit. This gravitational tug-of-war balances the planet’s natural forward motion and prevents it from flying off into space. The centripetal force maintains the delicate equilibrium between the planet’s inertia and the Sun’s gravitational force, ensuring that the planet remains in a stable and predictable elliptical orbit around the Sun.

### Globe of Death

Inside the captivating Globe of Death, a skilled biker performs daring feats by riding in circular loops. The centripetal force is the unseen hero here, acting as the invisible hand that keeps the biker firmly inside the globe. As the biker accelerates and maintains speed, the centripetal force constantly pulls towards the center of the circular path. This force counteracts gravity‘s pull, allowing the biker to stay glued to the inner surface of the globe, defying the natural urge to fall downwards. The combination of the biker’s momentum and the centripetal force creates a thrilling spectacle, showcasing the delicate balance between velocity and the centripetal force that enables the biker’s gravity-defying performance.

### Roller coaster

As a roller coaster hurtles along its twists and turns, riders experience a unique sensation of being pushed to the side. This sensation is a direct result of centripetal force at play. The coaster’s rapid changes in direction require a force that redirects the passengers’ motion inward, toward the center of each curve. This inward force is the centripetal force, and it ensures that the riders stay securely on the track despite the coaster’s dramatic maneuvers. Without the centripetal force, riders would continue moving in a straight line, causing them to fly off the track due to inertia. The interplay between the coaster’s speed, the track’s shape, and the centripetal force creates an exhilarating and gravity-defying experience that leaves riders thrilled and amazed.

### Roulette

In a game of roulette, the spinning wheel adds an element of excitement as the small ball travels along a curving path. This captivating motion is the result of centripetal force, a pulling force directed towards the center of the spinning wheel. This force prevents the ball from moving in a straight line and ensures that it stays connected to the wheel’s circular motion. As the wheel slows down, the centripetal force decreases, allowing the ball to settle into one of the numbered pockets. The interplay between the ball’s inertia, the wheel’s rotation, and the centripetal force creates the dynamic and suspenseful game that casino-goers enjoy.

## Equation

The centripetal force equation, represented as Fc = (m × v2) ÷ r, determines the force necessary to maintain an object’s motion in a circular trajectory. It is denoted by Fc and represents the inward force required to keep the object on its circular path, thus preventing it from moving in a straight line tangent to the circle. The variables m, v, and r correspond to the mass of the object, its velocity, and the radius of the circular path, respectively.

### Practice problems

#### Problem #1

A race car weighing 800 kg moves on a circular track with a velocity of 180 km/h. Determine the centripetal force required to keep the race car in its circular path. The radius of the circular track is 50 m.

Solution

Given data:

• Mass of the race car, m = 800 kg
• Velocity of the race car, v = 180 km/h = 50 m/s
• Centripetal force acting on the race car, Fc = ?
• Radius of the circular path, r = 50 m

Using the equation:

• Fc = (m × v2) ÷ r
• Fc = [800 × (50)2] ÷ 50
• Fc = (2 × 106) ÷ 50
• Fc = 40,000 N

Therefore, the centripetal force acting on the race car is 40,000 N.

#### Problem #2

A satellite with a mass of 500 kg orbits in a circular path with a velocity of 6000 m/s. What is the centripetal force acting on the satellite? The radius of the circular orbit is 2000 km.

Solution

Given data:

• Mass of the satellite, m = 500 kg
• Velocity of the satellite, v = 6000 m/s
• Centripetal force acting on the satellite, Fc = ?
• Radius of the circular orbit, r = 10 m

Using the equation:

• Fc = (m × v2) ÷ r
• Fc = [500 × (6000)2] ÷ 2000
• Fc = (18 × 109) ÷ 2000
• Fc = 9 × 106 N

Therefore, the centripetal force acting on the satellite is 9 × 106 N.

#### Problem #3

A roller coaster train weighing 600 kg moves through a circular loop with a radius of 30 m. At a velocity of 20 m/s, what is the centripetal force acting on the roller coaster car to maintain its circular motion?

Solution

Given data:

• Mass of the roller coaster train, m = 600 kg
• Radius of the circular loop, r = 30 m
• Velocity of the roller coaster train, v = 20 m/s
• Centripetal force acting on the roller coaster train, Fc = ?

Using the equation:

• Fc = (m × v2) ÷ r
• Fc = [600 × (20)2] ÷ 20
• Fc = (240 × 103) ÷ 20
• Fc = 12,000 N

Therefore, the centripetal force acting on the roller coaster train is 12,000 N.

#### Problem #4

A ball of mass 0.2 kg swings in a circular path with a radius of 0.5 m. It has a velocity of 4 m/s. Calculate the centripetal force acting on the ball, considering it is attached to a string.

Solution

Given data:

• Mass of the ball, m = 0.2 kg
• Radius of the circular path, r = 0.5 m
• Velocity of the ball, v = 4 m/s
• Centripetal force acting on the ball, Fc = ?

Using the equation:

• Fc = (m × v2) ÷ r
• Fc = [0.2 × (4)2] ÷ 0.5
• Fc = 3.2 ÷ 0.5
• Fc = 6.4 N

Therefore, the centripetal force acting on the ball is 6.4 N.