Question

1. Using words and a mathematical expression, describe the relationship between force and mass in uniform circular motion.2. Using words and a mathematical expression, describe the relationship between force and velocity in uniform circular motion.3. Using words and a mathematical expression, describe the relationship between force and radius in uniform circular motion.4. Combine the three relationships above to create one relationship for force, mass, velocity, and radius.5. How would you convert this expression into an equation?6. What is the constant of proportionality for this equation? Explain.7. How could such an equation be used?

Answer 1

1.

In a uniform circular motion, the object is kept along a circular path by a net force acting on it, called centripetal force, which always acts towards the centre of the circular trajectory.

Mathematically, we can write

$F \propto m$

where F is the centripetal force

m is the mass of the object

As we see from the equation, F (the force) is directly proportional to m (the mass).

2.

The relationship between force (F) and velocity (v) in a uniform circular motion is

$F \propto v^2$

where

F is the force

v is the magnitude of the velocity

so, we see that the force is proportional to the square of the magnitude of the velocity, $v^2$.

Let's also keep in mind that velocity is a vector, so it consists of a direction as well. In a circular motion, the direction of the velocity is tangential to the circular trajectory, while the centripetal force is radial, towards the centre of the circle (so, it is perpendicular to the velocity)

3.

The relationship between the force (F) and the radius (r) in a uniform circular is

$F\propto \frac{1}{r}$

and this means that the force is inversely proportional to the radius of the trajectory, r.

4.

Combining the three expressions that we wrote previously, we find a relationship between the centripetal force and all the other 3 quantities:

$F=m\frac{v^2}{r}$

where

F is the force

m is the mass of the object

v is the tangential speed of the object

r is the radius of the orbit

5.

An equation can be derived if we consider, for instance, the motion of a planet around a star. In that case, the centripetal force is provided by the gravitational attraction between the star and the planet. So we can rewrite F as

$\frac{GMm}{r^2}=m\frac{v^2}{r}$

where

G is the gravitational constant

M is the mass of the star

m is the mass of the planet in circular orbit around the star

r is the orbital radius of the planet's orbit

v is the speed of the planet

6.

First of all we notice that one term 'm' and one term 'r' can be simplified by the previous equation:

$\frac{GM}{r}=v^2$

Which can be rewritten as

$v^2 r=GM$

So in this case the product (GM) represents a constant term, and so the term $v^2 r$ is constant for every planet orbiting the same star.

7.

The equation can be used in several ways. For instance, it is possible to calculate the orbital speed of a planet revolving around the Sun, by re-arranging the equation as:

$v=\sqrt{\frac{GM}{r}}$

where

G is the gravitational constant

M is the mass of the star

r is the orbital radius of the planet's orbit

Answer 2

This question involves the concepts of uniform circular motion and centripetal force.

The answers to the questions are given below one by one:

1.

The force exerted on a particle due to uniform circular motion is known as the centripetal force. It acts towards the center of the circular path. This centripetal force is directly proportional to the mass of the object.

$F\ \alpha\ m$

2.

The centripetal force is directly proportional to the square of the velocity of the object.

$F\ \alpha\ v^2$

3.

The centripetal force is inversely proportional to the radius of the circular path.

$F\ \alpha\ \frac{1}{r}$

4.

Combining all the relationships given above, we get:

$F\ \alpha\ \frac{mv^2}{r}$

5.

The equation of the centripetal force formed through this relationship is:

$F=\frac{mv^2}{r}$

6.

The constant of proportionality in this relation is the mass (m). Because the mass of an object remains constant everywhere in the universe.

7.

This expression can be used to find out any one of the four variables in a uniform circular motion, when the other three are known.

For example: When the mass of object, radius of the path, and the centripetal force are known, the speed of the object can be found using this formula.

Also, the centripetal force often acts like some other force, such as tension in the rope, gravitational force between objects in circular motion. Hence, this formula can also be used to find such forces.

Learn more about centripetal force here:

brainly.com/question/11324711?referrer=searchResults

The attached picture shows the centripetal force.