6.3 Centripetal Force

The unitless radians are discarded in order to get the correct units.

The centripetal acceleration is 472,000 times stronger than.

Ultracentrifuges are called such because of the high number of them.
The time needed to cause the destruction of blood cells or other materials is greatly reduced by the large accelerations.

The second law of motion states that a net external force is needed to cause any acceleration.

A centripetal acceleration can be caused by a net external force.
The forces involved in circular motion will be considered in Centripetal Force.

You can learn about position, velocity and acceleration.

The ladybug can be moved by setting the position, speed or acceleration.
To analyze the behavior, choose linear, circular or elliptical motion.

A centripetal or radial acceleration can be caused by any force or combination of forces.

The force of Earth's gravity on the Moon, the tension in the rope on a tether ball, and the force of a banked roadway on a car are just a few examples.

The direction of a centripetal force is the same as the direction of centripetal acceleration.

The second law of motion states that net force is mass times acceleration.
The centripetal acceleration is used for uniform circular motion.

You can use whichever expression is more convenient.

Centripetal force is always pointing to the center of curvature because it is always parallel to the path.

A tight curve is caused by a large centripetal force for a given mass.

The centripetal force is equal to the frictional force.

Centripetal force causes a circular motion.
The second curve has the same as the first, but a larger produces a smaller one.

The forces acting on the car are shown in Figure 6.12.

The centripetal force on the car is the only horizontal force acting on the car, and so the Friction is to the left.
We know that the maximum static friction is, where is the static coefficient of friction and N is the normal force.
The normal force is equal to the car's weight on the ground.

There is a relationship between centripetal force and the coefficients of friction.

The first expression in because and are given could be used to solve part a.

In part (b) the coefficients of friction are much smaller than they are between tires and roads.
The car will still negotiate the curve if the coefficients are greater than 0.13 because static friction is a responsive force, being able to assume a value less than but no more.
The safe speed is less than 25 m/s if the coefficient of friction is less.
In this example, mass cancels, implying that it doesn't matter how heavy the car is to negotiate the turn.
Mass cancels because the force is assumed to be proportional to the mass.
The normal force on the road would be reduced if it were banked.

The car is moving away from the ground.

The car turning in a circular path is due to the centripetal force between the tires and the road.
The car will move in a larger-radius curve if there is a minimum coefficient of friction.

You can take the curve quicker if the angle is greater.

steeply banked curves are often found on race tracks for bikes and cars.

We will come up with an expression for an ideally banked curve.

The weight of the car and the centripetal force of the normal force N must be equal in the horizontal and vertical directions.

The vertical and horizontal directions are the most convenient ways to consider components in cases where forces are not parallel.

The diagram in Figure 6.13 shows a car on a banked curve.

The net external force will equal the centripetal force if the angle is ideal.
The normal force of the road and the weight of the car are the only external forces acting on the car.

The coordinate system we use is based on the horizontal force.

The net vertical force is zero because the car does not leave the surface of the road.

The vertical component of the normal force is, and the only other vertical force is the car's weight.

We can combine the last two equations to get an expression we want.

The expression depends on and can be understood.

The roads must be steeply banked for high speeds.
Friction helps because it allows you to take the curve at different speeds.
It doesn't depend on the mass of the vehicle.

The car is moving away from the banked curve.

The Daytona International Speedway in Florida is one of the test tracks that have steeply banked curves.

The curves can be taken at very high speed with the aid of this banking.

If the road is open, calculate the speed at which a 100 m radius curve should be driven.

We note that the expression for the ideal angle of a banked curve except for speed is known, so we need only rearrange it so that speed appears on the left-hand side and then substitute known quantities.

This is close to 165 km/h, which is consistent with a very steeply banked curve.

A vehicle can take the curve at higher speeds.

There are a number of interesting situations in which centripetal force is involved that can be calculated in the same way as those in the preceding examples.

A friend or relative can swing a golf club or tennis racquet.

The centripetal acceleration of the end of the club or racquet can be measured.
You can do this in slow motion.

Move the sun, earth, moon and space station to see how they affect each other.

Most people agree that when taking off in a jet it feels as if you are being pushed back into the seat.

A physicist says that you tend to remain stationary while the seat pushes forward on you, and there is no force on you.
A more common experience is when you make a tight curve in your car.
You feel like you're being thrown toward the left relative to the car.
A physicist would say that you are going in a straight line but the car moves to the right and there is no force on you to the left.

The force is a result of the use of the car as a reference.

There is no force to the left on the driver.
There is a force to the right on the car.

The frames of reference used can be used to reconcile the points of view.

People are in a car.

Passengers use the car as a reference point, while physicists use Earth.

Earth is almost an insturment frame of reference, one in which all forces are real and have an identifiable physical origin.
The driver is actually going to the right and there is nothing real pushing them left.

We can take a mental ride on a merry-go-round.

You take the merry-go-round to be your reference point.
To counteract the force, you must hang on tightly.
There is no force trying to throw you off.
You have to hang on to make yourself go in a circle because otherwise you would go straight off the merry-go-round.

The rider's motion in the rotating frame of reference is explained by the fictitious force.

A force is needed to cause a circular path.

A sample is spun very quickly by a centrifuge.

From the rotating frame of reference, you can see the force that throws particles out.

The test tube is forced in a circular path by a centripetal force while the particles are carried along a line tangent to the circle.

Centrifuges use inertia.

inertia carries the particles away from the center of rotation.
The speed of thecentrifugation quickens.
The centripetal force needed to make the particles move in a circle of constant radius will come from the particles coming into contact with the test tube walls.

If something moves in a frame of reference that rotates, what happens?

The ball follows a straight path relative to Earth and a curved path to the right on the merry-go-round's surface.
A person standing next to the merry-go-round sees the ball move straight and the merry-go-round rotating underneath it.
The Coriolis force can be used to explain why objects follow curved paths and can be applied to non-inertial frames of reference.

Looking down on the merry-go-round, we can see that a ball goes straight toward the edge.

The person slides the ball towards point A.
The shaded positions of the points are shown in the time that the ball follows the curved path in the rotating frame and the straight path in Earth's frame.

Earth has been considered an insturment with little or no worry about effects due to its rotation.

There are effects in the rotation of weather systems.
The consequences of Earth's rotation can be understood by analogy with the merry-go-round.
Any motion in the northern hemisphere experiences a Coriolis force to the right.
The force is to the left in the southern hemisphere.
For large-scale motions, such as wind patterns, the Coriolis force can have significant effects.

The hurricanes in the northern hemisphere are caused by the Coriolis force, while the tropical cyclones in the southern hemisphere are caused by a different force.

Tropical storms are characterized by low pressure centers, strong winds, and heavy rains.
Tropical cyclones have low pressures and air flows toward them.
The winds flow toward the center of the tropical storm or low-pressure weather system.
In the northern hemisphere, the inward winds are directed to the right and produce a counterclockwise circulation at the surface for low-pressure zones.
Low pressure at the surface is associated with rising air, which also produces cooling and cloud formation, making low-pressure patterns visible from space.
The wind circulation around high-pressure zones is clockwise in the northern hemisphere, but it is less visible because of the sinking air.

The path of a ball on a merry-go-round can be explained by the rotation of the system underneath.

The Coriolis force must be invented to explain the curved path when non-inertial frames are used.
There isn't a physical source for these forces.
inertia explains the path and no force is found to be without an identifiable source.
The simplest and truest view is the one in which all forces have real origins and explanations.