The force exerted on a car by a circular ramp is dependent on the speed at which the car travels along the ramp and the radius of the ramp. If the car travels at a slower speed or on a smaller radius, then more force is exerted on the car.
The math involved in calculating this force is complex, but your friendly physics calculator can do it for you. So, let’s get started!
If we imagine a circle with a width of 100 meters and a height of 100 meters, we can use physics to calculate how hard it would be to climb out using only your hands. First, we will need to figure out the force pulling you down into the circle.
This article will explain how to calculate this force and what factors influence it.
Definition of circular motion
Circular motion is the term used to describe the motion of a car or other vehicle that is moving in a circular path. A circular path is the route that a vehicle takes as it travels, typically characterized by a starting and ending point and a continuous line connecting the two points.
Most commonly, vehicles travel in elliptical paths, where the difference between the axes of rotation is not equal. However, any path that can be described as a continuous curve can be considered a circle.
When analyzing the physics of circular motion, it is important to distinguish between linear and angular velocity. Linear velocity refers to how far something moves in a given time period; for example, how far down a car goes in one minute. Angular velocity refers to how much something rotates in a given time period; for example, how much its wheels turn during that time.
Relationship between velocity and radius
When a vehicle moves in a circle, its velocity is determined by the radius of the circle it travels in. The greater the radius, the slower the vehicle will travel; likewise, the smaller the radius, the faster the vehicle will travel.
This is because there is more linear distance traveled per rotation of the circle. The greater the radius, the less linear distance that is traveled per rotation.
This fact can be proven with mathematics, where r equals the radius and θ equals 360 degrees or one full rotation. The equation for velocity is v=rθ, so reducing both sides to r gives you v=vθ, or velocity equals frequency times length.
Therefore, if you increase length (radius), then frequency (velocity) must decrease to maintain consistency.
Relationship between acceleration and radius
Now let’s return to the car traveling in a circular path at constant speed. What about the net force on the car?
Acceleration is one factor that affects the net force on a vehicle. If the radius of the circle changes, then the acceleration must also change in order for the vehicle to maintain constant speed.
How does this happen? It’s all related to something called angular velocity. Angular velocity is how many degrees per unit time a object turns.
Imagine that there is a clock inside of the car and it takes one hour for it to complete one full rotation. Then the angular velocity would be one degree per hour, or 360 degrees per day.
If you change either of these numbers, then the other must also change in order for the car to still maintain constant speed.
Centripetal force equation
Now let’s consider a car traveling in a circular path at constant speed. Since the car is moving, we have to consider its motion relative to the ground.
We can use a similar equation for the centripetal force, but now we must account for the speed of the car as well as the radius of the circle it is traveling in.
The faster the car travels, the greater the required force to keep it in a circle. This is because there is less time for it to alter its direction. The required force depends on how much time it has to change direction, so its magnitude depends on speed as well.
Consider two identical cars traveling in circles of different radii but with the same speed. The larger circle requires more force than the smaller circle, so there must be more centripetal force acting on it.
Car traveling in a circular path at constant speed
Now let’s consider a car traveling in a circular path at constant speed. Since the car is traveling in a circular path, the acceleration is arithmetic.
Since the car is traveling at constant speed, the acceleration is steady. Because the car is accelerating in a circle, this type of acceleration is called radial acceleration.
Given these specifications, we can use the following formula to determine the net force exerted on the car:
This force is what pushes you back into your seat! You might notice that this force has both a magnitude and direction; it points outward from the center of the circle toward the outside of the circle.
Like with linear acceleration, you can also express radial acceleration as a multiplier of normal gravity (ng). Radial acceleration can be written as ar=arng, where r=rng.
What happens to the centripetal force when the object moves at a non-constant speed?
When the car speed is not constant, the net force on the car is not zero. There is still a force acting on the car, but it is not directed along the circular path.
To understand this concept, think about how you walk down a hill. When you take a step forward, you experience a force pushing you back. This force is gravity acting on you, pulling you down the hill.
But since you are taking a step forward, your body does not move in a straight line down the hill. You are moving in a circle with your foot as its axis. Because of this motion, your body does not experience a net force acting on it that pulls you down the hill. You stay upright and moving in a circle!
The same thing happens when you drive your car around a curve. The net force on the car due to angular acceleration is zero.
What happens to the centripetal force when the object moves in a non-circular path?
If the object does not travel in a circular path, then the net force on the object is not zero. There is always a force acting on an object, even if it is in motion and not touching anything else.
If the object moves in a straight line, then the only force acting on it is the ground exerting a force up on it. If it moves in a circle, then it is also being pulled toward the center of the circle.
To find out how much force is pulling an object toward the center of a circle, you have to take its speed and divide by two times its radius. This gives you what mathematicians call its angular velocity. Then you can find out how much force that exerts by using something called newton’s law of universal gravitation.
Applications of centripetal force
Centripetal force is applied in many situations. Every time you throw a ball, you give it a forward push as it spins, making it travel in a curved path.
This is done by applying a force towards the rotation of the ball. This force is called a centripetal force, as it directs the object toward the center of rotation.
In physics classes, students learn about special cases of centripetal forces. One such example is when an object moves in a circle with constant speed. In this case, the force acting on the object is exclusively its weight.
This is because, in physics terms, weight is defined as g times acceleration due to gravity where g is 9.8 m/s^2. Since acceleration due to gravity remains constant regardless of location, this stays constant for an object moving in a circle with constant speed.