Question

An object with mass \(m\) moves in a spiral orbit \(r=c \theta^{2}\) under a central force $$ \mathbf{F}(r, \theta)=f(r)(\cos \theta \mathbf{i}+\sin \theta \mathbf{j}) $$ Find \(f\).

Short answer

Based on the given step-by-step solution, the central force function, \(f(r)\), which represents the force acting on an object of mass \(m\) moving in a spiral orbit defined by the equation \(r = c\theta^2\), is given by: $$ f(r) = \frac{2c}{m}\frac{d^2\theta}{dt^2} - \frac{4c\theta}{m}\left(\frac{d\theta}{dt}\right)^2 $$

Step-by-step solution

Express position and velocity in polar coordinates

In polar coordinates, the position vector of the object can be expressed as: $$ \mathbf{r} = r(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}) $$ Since \(r = c\theta^2\), we have: $$ \mathbf{r} = c\theta^2(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}) $$ Now, let's find the velocity vector \(\mathbf{v}\) by taking the time derivative of the position vector \(\mathbf{r}\): $$ \mathbf{v} = \frac{d\mathbf{r}}{dt} = \frac{d(c\theta^2)}{dt}(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}) + c\theta^2\frac{d(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j})}{dt} $$ Using the chain rule, we get: $$ \mathbf{v} = 2c\theta\frac{d\theta}{dt}(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}) - c\theta^2(\sin{\theta}\mathbf{i}-\cos{\theta}\mathbf{j})\frac{d\theta}{dt} $$

Express the acceleration in polar coordinates

Now, we need to find the acceleration \(\mathbf{a}\). Let's take the time derivative of the velocity vector \(\mathbf{v}\): $$ \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d(2c\theta\frac{d\theta}{dt}(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}))}{dt} - \frac{d(c\theta^2(\sin{\theta}\mathbf{i}-\cos{\theta}\mathbf{j})\frac{d\theta}{dt})}{dt} $$ After applying the chain rule and simplifying, we get: $$ \mathbf{a} = [2c\frac{d^2\theta}{dt^2}(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}) - 2c\theta\frac{d\theta}{dt}(\sin{\theta}\mathbf{i}-\cos{\theta}\mathbf{j})\frac{d\theta}{dt}] - [2c\theta\frac{d^2\theta}{dt^2}(\sin{\theta}\mathbf{i}-\cos{\theta}\mathbf{j}) + 4c\theta^2\frac{d\theta}{dt}(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j})\frac{d\theta}{dt}] $$

Apply Newton's Second Law to find the central force

Now, let's apply Newton's second law, \(\mathbf{F} = m\mathbf{a}\), and write down the equation for the central force \(\mathbf{F}(r, \theta)\): $$ \mathbf{F}(r, \theta) = m\mathbf{a} = mf(r)(\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}) $$ Since \(\mathbf{a}\) and \(\mathbf{F}(r, \theta)\) have the same direction, we can equate their coefficients: $$ mf(r) = 2c\frac{d^2\theta}{dt^2} - 4c\theta\left(\frac{d\theta}{dt}\right)^2 $$ We can now solve for the function \(f(r)\) by dividing by \(m\): $$ f(r) = \frac{2c}{m}\frac{d^2\theta}{dt^2} - \frac{4c\theta}{m}\left(\frac{d\theta}{dt}\right)^2 $$ This is the expression for the central force acting on the object moving in a spiral orbit.

Key concepts

Polar Coordinates

Polar coordinates provide a two-dimensional coordinate system where each point is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the distance from the pole is denoted as \(r\). Meanwhile, the angle, denoted as \(\theta\), is measured from a fixed direction. This system is useful in problems involving rotational symmetry and contributes significantly to simplifying calculations involving spirals or circular paths.

The position of an object in polar coordinates is given by \(\mathbf{r} = r(\cos \theta \mathbf{i} + \sin \theta \mathbf{j})\). Here, \(r\) is the radial distance, and \(\theta\) is the angular coordinate. When the position is described as \(r = c\theta^2\), it implies that the object travels in a spiral path, as \(\theta\) changes. This description is vital when determining the components of velocity and acceleration, which are necessary for further analysis using other physical laws.

Newton's Second Law

Newton's Second Law establishes the relationship between an object's mass, acceleration, and the applied force, formulated as \( \mathbf{F} = m\mathbf{a} \). This law is critical because it connects force and acceleration through mass, allowing us to predict how an object will move under the influence of various forces.

When applied to the exercise, this law helps us relate the central force \(\mathbf{F}(r, \theta)\) on the object to its mass \(m\) and its acceleration \(\mathbf{a}\), which was computed in polar coordinates. By asserting that the product of mass and the derived acceleration equals the central force in polar form \(mf(r)(\cos \theta \mathbf{i} + \sin \theta \mathbf{j})\), we can deduce a formula for \(f(r)\), the function describing the central force's magnitude depending on \(r\). This step is vital for solving the exercise.

Chain Rule

The chain rule is a fundamental principle in calculus used to find the derivative of a composite function. In simple terms, it helps us differentiate a function that is nested within another function. This rule is expressed mathematically as:

• \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \)

This concept is incredibly important when dealing with velocity and acceleration in polar coordinates, where the position of an object is described as \(r = c\theta^2\).

For the given problem, as the object moves in a spiral orbit, both \(r\) and \(\theta\) are functions of time, \(t\). The chain rule is used to compute derivatives of these functions over time, which are essential for determining the components of the velocity and acceleration vectors. This enables us to express the time change of quantities like \(\theta\) in terms of their rates—helping us, for example, find \(\frac{d^2\theta}{dt^2}\). Mastery of the chain rule aids in solving many complex physics problems involving time-dependent variables.

Differential Equations

Differential equations are equations involving an unknown function and its derivatives. They are crucial because they describe a variety of phenomena where quantities change over time or space, such as motion, heat, or wave propagation.

In the context of the exercise, differential equations emerge when describing the motion of the object in the spiral orbit. We've derived expressions involving derivatives of \(\theta\), such as \( \frac{d^2\theta}{dt^2} \), using the concepts like velocity and acceleration in polar coordinates. Solving these equations helps us predict how the angle \(\theta\) varies with time.

To find the central force function \(f(r)\), we equated mass times acceleration to the force functions and solved for \(f(r)\) involving derivatives. This is a typical application of differential equations, as they allow us to find relationships among the physical quantities and their rates of change. Understanding differential equations is directly linked to addressing this problem.