Uniform Circular Motion
We have seen that if the is found to be perpendicular to an object’s motion then it can’t do any on the object. Therefore, the net force will only change the object’s direction of motion, change it’s ) and the object must maintain a . The object will undergo , in which case we sometimes refer to the net force that points toward the center of the circular motion as the , but this is just a naming convention. The centripetal force is not a new kind of force, rather the centripetal force is provided by one of the forces we already know about, or a combination of them. For example, the centripetal force that keeps a satellite in orbit is just and for a ball swinging on a string in the string provides the .
For both the ball and the satellite the points at 90° to the object’s motion so it can do no , thus it cannot change the of the object, which means it cannot change the of the object. How do we mesh this with , which says that objects with a net force must experience ? We just have to remember that acceleration is change per time and velocity includes and direction. Therfore, the constantly changing direction of constitutes a constantly changing velocity, and thus a constant, so all is good. Due to , we know that the points toward the center of the circular motion because that is where the points. As a result, that acceleration is called the . If the drops to zero (string breaks) the acceleration must become zero and the ball will continue off at the same speed in whatever direction it was going when the net force became zero (diagram on right above).
Centripetal Force and Acceleration
The size of the acceleration experienced by an object undergoing with radius at is:
Combined with we can find the size of the , which again is just the during :
Everyday Example: Rounding a Curve
What is the maximum that a car can have while rounding a curve with radius of 75 m without skidding? Assume the between tire rubber and the asphalt road is 0.7
First, we recognize that as the car rounds the curve at the must point toward the center of the curve and have the value:
Next we recognize that the only force available to act on the car in the horizontal direction (toward the center of the curve) is , so the in the horizontal direction must be just the frictional force:
We want to know the maximum speed to take the curve without slipping, so we need to use the maximum that can be applied before slipping:
Notice that we have used static friction even though the car is moving because we are solving the case when the tires are still rolling and not yet sliding. Kinetic friction would be used if the tires were sliding.
For a typical car on flat ground the will be equal to the of the car:
Then we cancel the from both sides of the equation and solve for :
Inserting our values for friction coefficient, g, and radius:
When you stand on a scale and you are not in , then the may not be equal to your and the weight measurement provided by the scale will be incorrect. For example, if you stand on a scale in an elevator as it begins to move upward, the scale will read a weight that is too large. As the elevator starts up, your motion changes from still to moving upward, so you must have an upward and you must not be in equilibrium. The from the scale must be larger than your weight, so the scale will read a value larger than your weight.
In similar fashion, if you stand on a scale in an elevator as it begins to move downward the scale will read a that is too small. As the elevator starts down, your motion changes from still to moving downward, so you must not be in , rather you have a downward . The from the scale must be less than your weight.
Taking the elevator example to the extreme, if you try to stand on a scale while you are in , the scale will be falling with the same as you. The scale will not be providing a to hold you up, so it will read your as zero. We might say you are weightless. However, your weight is certainly not zero because weight is just another name for the, which is definitely acting on you while you . Maybe normal-force-less would be a more accurate, but also less convenient term than weightless.
We often refer to astronauts in orbit as weightless, however we know the must be acting on them in order to cause the required for them to move in a circular orbit. Therefore, they are not actually weightless. The astronauts feel weightless because they are in along with everything else around them. A scale in the shuttle would not read their weight because it would not need to supply a to cancel their weight because both the scale and the astronaut are in free fall toward Earth. The only reason they don’t actually fall to the ground is that they are also moving so fast to their downward that by the time they would have hit the ground, they have moved sufficiently far to the side that they end up falling around the Earth instead of into it.
Everyday Example: Orbital Velocity
How fast does an object need to be moving in order to around the Earth (remain in orbit)? We can answer that question by setting the equal to the , given by Newton’s ( = mg is only valid for object near Earth’s surface, remember):
Recognizing that is the in this case, and that is the Earth’s and is the orbiting object’s mass:
Cancelling and one factor of from both sides and solving for :
We see that the necessary orbit speed depends on the radius of the orbit. Let’s say we want a low-Earth orbit at an altitude of 2000 km, or . The radius of the orbit is that altitude plus the Earth’s radius of to get or . Inserting that total radius and the gravitational constant, , and the Earth’s mass: :
Use this simulation to play with the velocities of these planets in order to create stable orbits around the sun.
the total amount of remaining unbalanced force on an object
A quantity representing the effect of applying a force to an object or system while it moves some distance.
energy which a body possesses by virtue of being in motion, energy stored by an object in motion
not changing, having the same value within a specified interval of time, space, or other physical variable
distance traveled per unit time
motion of an object traveling at a constant speed on a circular path
a name given to the component of the net force acting perpendicular to an objects motion and causing to take a circular path
the force that is provided by an object in response to being pulled tight by forces acting from opposite ends, typically in reference to a rope, cable or wire
the acceleration experienced by an object is equal to the net force on the object divided my the object's mass
the change in velocity per unit time, the slope of a velocity vs. time graph
a quantity of speed with a defined direction, the change in speed per unit time, the slope of the position vs. time graph
component of the acceleration directed toward the center of a circular motion
coefficient describing the combined roughness of two surfaces and serving as the proportionality constant between friction force and normal force
a force that acts on surfaces in opposition to sliding motion between the surfaces
a force that resists the tenancy of surfaces to slide across one another due to a force(s) being applied to one or both of the surfaces
the outward force supplied by an object in response to being compressed from opposite directions, typically in reference to solid objects.
the force of gravity on on object, typically in reference to the force of gravity caused by Earth or another celestial body
a measurement of the amount of matter in an object made by determining its resistance to changes in motion (inertial mass) or the force of gravity applied to it by another known mass from a known distance (gravitational mass). The gravitational mass and an inertial mass appear equal.
a state of having no unbalanced forces or torques
the motion experienced by an object when gravity is the only force acting on the object.
attraction between two objects due to their mass as described by Newton's Universal Law of Gravitation
at an angle of 90° to a given line, plane, or surface
every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers