Question

List some formulas that occur in applications. Solve each formula for the indicated variable. $$ \text { Mechanics } F=\frac{m v^{2}}{R} \text { for } R $$

Short answer

The formula solved for \( R \) is \( R = \frac{mv^2}{F} \).

Step-by-step solution

Understand the Formula

The given formula is from mechanics: \( F = \frac{mv^2}{R} \). This represents the force due to a mass \( m \) moving with velocity \( v \) in a circular path of radius \( R \). The force \( F \) is given by \( F \), but we need to solve this formula for \( R \).

Isolate the Desired Variable

To solve for \( R \), isolate \( R \) by manipulating the equation. Start by multiplying both sides of the equation by \( R \) to eliminate the denominator on the right side: \( F \cdot R = mv^2 \).

Solve for R

Next, divide both sides by \( F \) to isolate \( R \): \( R = \frac{mv^2}{F} \). This is the expression for the radius \( R \) in terms of the other variables \( m \), \( v \), and \( F \).

Key concepts

solve for variable

In mechanics and many other fields, being able to solve for a specific variable is crucial. This means manipulating an equation to isolate a particular variable on one side. For example, in the exercise provided, we began with the force equation: \( F = \frac{mv^2}{R} \), where we needed to solve for \( R \).

Here are the steps:

• Identify the variable to solve for (in this case, \( R \)).
• Use algebraic operations to manipulate the equation.
• Multiply both sides by \( R \) to get rid of the denominator.
• Then, divide both sides by \( F \) to isolate \( R \).

These steps help in making the variable of interest stand alone. This technique can be applied to numerous equations across different fields.

force equation

In physics, the force equation is used to calculate the force exerted by an object. Specifically, the equation provided in the exercise \( F = \frac{mv^2}{R} \) describes the centripetal force, which is the force required to keep an object moving in a circular path.

Several key aspects of this equation are:

• \textbf{Force (\(F\))}: The push or pull on an object.
• \textbf{Mass (\(m\))}: The amount of matter in the object.
• \textbf{Velocity (\(v\))}: The speed and direction of the object's movement.
• \textbf{Radius (\(R\))}: The distance from the center of the circular path to the object.

Understanding these variables allows us to calculate the required force for an object to travel in a circular path with a given mass and velocity.

circular motion

Circular motion occurs when an object moves along a circular path. The centripetal force equation, \( F = \frac{mv^2}{R} \), is essential for understanding this motion.

Key concepts include:

• \textbf{Centripetal force}: This inward force keeps the object moving in a circle and points towards the center of rotation.
• \textbf{Centripetal acceleration}: The rate at which the object's velocity changes direction as it travels around the circle.

For an object to maintain circular motion, there must be a force acting towards the center of the circle. This is what keeps the object from flying off in a straight line due to inertia. The centripetal force formula captures the relationship between mass, velocity, radius, and force, making it easier to predict and measure these physical phenomena.