Relationship between centripetal force mass and radius

Uniform circular motion

When an object is traveling in a circular path, centripetal force is but you were creating a balance between two forces - one real and the center of the circle is represented by the variable r, or radius. How to Calculate Centripetal Force. Centripetal force is easily calculated as long as you know the mass. And that tells us if the velocity speeds up the force will be stronger and the radius well be smaller. Only if the (linear) velocity remains the same. An equation expresses a mathematical relationship between the quantities present in that For a constant mass and radius, the Fnet is proportional to the speed2. . Determine the centripetal force acting upon a kg child who makes

This can be solved to get the angle: In the x direction there's just the one force, the horizontal component of the tension, which we'll set equal to the mass times the centripetal acceleration: We know mass and tension and the angle, but we have to be careful with r, because it is not simply the length of the rope. It is the horizontal component of the 1. Rearranging this to solve for the speed gives: Example 2 - Identical objects on a turntable, different distances from the center.

Let's not worry about doing a full analysis with numbers; instead, let's draw the free-body diagram, and then see if we can understand why the outer objects get thrown off the turntable at a lower rotational speed than objects closer to the center. In this case, the free-body diagram has three forces, the force of gravity, the normal force, and a frictional force.

The friction here is static friction, because even though the objects are moving, they are not moving relative to the turntable. If there is no relative motion, you have static friction. The frictional force also points towards the center; the frictional force acts to oppose any relative motion, and the object has a tendency to go in a straight line which, relative to the turntable, would carry it away from the center.

So, a static frictional force points in towards the center. Summing forces in the y-direction tells us that the normal force is equal in magnitude to the weight.

Mathematics of Circular Motion

In the x-direction, the only force there is is the frictional force. The maximum possible value of the static force of friction is As the velocity increases, the frictional force has to increase to provide the necessary force required to keep the object spinning in a circle.

If we continue to increase the rotation rate of the turntable, thereby increasing the speed of an object sitting on it, at some point the frictional force won't be large enough to keep the object traveling in a circle, and the object will move towards the outside of the turntable and fall off.

Why does this happen to the outer objects first? More force is needed for the outer objects at a given rotation rate, and they'll reach the maximum frictional force limit before the inner objects will.

For instance, the equation for Newton's second law identifies how acceleration is related to the net force and the mass of an object. The relationship expressed by the equation is that the acceleration of an object is directly proportional to the net force acting upon it. In other words, the bigger the net force value is, the bigger that the acceleration value will be. As net force increases, the acceleration increases. In fact, if the net force were increased by a factor of 2, the equation would predict that the acceleration would increase by a factor of 2.

Similarly, if the net force were decreased by a factor of 2, the equation would predict that the acceleration would decrease by a factor of 2.

How to Find the Centripetal Force With the Radius, Mass & Constant Speed : Physics & Math

Newton's second law equation also reveals the relationship between acceleration and mass. According to the equation, the acceleration of an object is inversely proportional to mass of the object.

Centripetal force

In other words, the bigger the mass value is, the smaller that the acceleration value will be. As mass increases, the acceleration decreases. In fact, if the mass were increased by a factor of 2, the equation would predict that the acceleration would decrease by a factor of 2. Similarly, if the mass were decreased by a factor of 2, the equation would predict that the acceleration would increase by a factor of 2.

As mentioned previously, equations allow for predictions to be made about the affect of an alteration of one quantity on a second quantity.

Since the Newton's second law equation shows three quantities, each raised to the first power, the predictive ability of the equation is rather straightforward. The predictive ability of an equation becomes more complicated when one of the quantities included in the equation is raised to a power.

Mathematics of Circular Motion

For instance, consider the following equation relating the net force Fnet to the speed v of an object moving in uniform circular motion. This equation shows that the net force required for an object to move in a circle is directly proportional to the square of the speed of the object. For a constant mass and radius, the Fnet is proportional to the speed2. The factor by which the net force is altered is the square of the factor by which the speed is altered.

Subsequently, if the speed of the object is doubled, the net force required for that object's circular motion is quadrupled.