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Chapter 2: Problem 49

The position of a particle as a function of time is \(\vec{r}=4 \sin 2 \pi t \hat{i}+4 \cos 2 \pi t \hat{j}\) (where \(t\) is time in second). Path of this particle will be (A) an ellipse (B) a hyperbola (C) a circle (D) any other curved path

Short Answer

Expert verified
The path of the particle is a circle with a radius of 4 units.

Step by step solution

01

Identify the individual components of the function

In the given function, \( r(t) = 4\sin 2\pi t \hat{i} + 4\cos 2\pi t \hat{j} \), the \( \hat{i} \) and \( \hat{j} \) are the unit vectors in the x and y directions respectively. So, the \(4\sin 2\pi t\ \) and \(4 \cos 2\pi t\ \) represents the x and y coordinates of the position of the particle at any time 't' respectively.
02

Analyze the form of the function

Here, from the standard form for the equation of a circle, \( x^2 + y^2= r^2 \), by substituting the functions of 't' we get, \( (4 \sin 2\pi t )^2 + (4\cos 2\pi t)^2 = 16 \). Simplifying it, we get, \(16(\sin^2 2\pi t + \cos^2 2\pi t ) = 16\). But \( \sin^2 2\pi t + \cos^2 2\pi t \) is a trigonometric identity which always equals to 1. So, we end up with the equation \(16 = 16\), which is true. Therefore, the given function represents a path which is a circle.
03

Identify the radius of the circle

After establishing the path as a circle, we can identify the radius of the circle which is 'r'. From the standard form \(x^2 + y^2 = r^2\), squaring the '4' coefficients in the given equation \(x = 4\sin 2\pi t \) and \(y = 4\cos 2\pi t\), we obtain the radius 'r' as '4'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are equations that are true for all values of the involved variables. These identities simplify the process of analyzing periodic functions, like those describing circular motion. A commonly used identity is \( \sin^2 \theta + \cos^2 \theta = 1 \), which holds for any angle \( \theta \). In our exercise, we applied this identity to combine the sine and cosine components of the position function into a unified expression, verifying the circular path of the particle.
  • \( \sin \theta \) and \( \cos \theta \) are the basic trigonometric functions representing a point on a unit circle.
  • The identity \( \sin^2 2\pi t + \cos^2 2\pi t = 1 \) illustrates that regardless of the value of \( t \), the sum of squared sine and cosine functions equals one.
  • This identity is foundational in trigonometry and helps simplify expressions involving trigonometric functions.
Understanding and utilizing trigonometric identities can make complex calculations easier by reducing expressions to more manageable forms.
Position Function
A position function describes the location of a particle or body at any given time. In this exercise, the position function is expressed as a vector function:\[\vec{r} = 4 \sin 2\pi t \ \hat{i} + 4 \cos 2\pi t \ \hat{j}\]This equation conveys both the x and y coordinates, defined by sine and cosine components respectively. Here鈥檚 how to interpret this function:
  • The term \( 4 \sin 2\pi t \) describes the x-coordinate of the position, combining the amplitude (4) with a periodic sine function.
  • Similarly, \( 4 \cos 2\pi t \) provides the y-coordinate, using cosine for periodicity in the vertical direction.
  • The combination of these terms into a vector expresses motion in two-dimensional space, which is essential when considering circular or elliptical trajectories.
A clear comprehension of position functions allows you to predict a particle鈥檚 path over time, telling a story of motion throughout a specific period.
Parametric Equations
Parametric equations are a set of equations that express a set of quantities as explicit functions of a common variable, often time. In the context of our exercise, \( x = 4 \sin 2\pi t \) and \( y = 4 \cos 2\pi t \) represent parametric equations. These are highly useful in describing curves and paths in the plane.
  • Each equation defines the coordinates in terms of an independent parameter, \( t \), rather than directly relating x and y.
  • This approach is particularly advantageous in modeling paths or trajectories that aren't easily expressed in standard Cartesian coordinates.
  • Using parametric equations, you gain a comprehensive view of motion, offering insights into the specific behavior of dynamic systems over time.
Ultimately, parametric equations give mathematicians and scientists a powerful tool to describe paths and surfaces in a precise and detailed manner.

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Most popular questions from this chapter

A particle is projected upwards with a velocity of \(100 \mathrm{~m} / \mathrm{s}\) at an angle of \(37^{\circ}\) with the vertical. The time when the particle will move perpendicular to its initial direction is \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}, \tan 53^{\circ}=4 / 3\right)\) (A) \(10 \mathrm{~s}\) (B) \(12.5 \mathrm{~s}\) (C) \(15 \mathrm{~s}\) (D) \(16 \mathrm{~s}\)

A projectile is fired with a velocity \(u\) at right angle to a slope, which is inclined at an angle \(\theta\) with the horizontal. The range of the projectile on the incline is (A) \(\frac{2 u^{2} \sin \theta}{g}\) (B) \(\frac{2 u^{2}}{g} \tan \theta \sec \theta\) (C) \(\frac{u^{2}}{g} \sin 2 \theta\) (D) \(\frac{2 u^{2}}{g} \tan \theta\)

A particle can be projected with a given speed in two possible ways so as to make it pass through a point at a distance \(r\) from the point of projection. The product of the times taken to reach this point in the two possible ways is then proportional to (A) \(r\) (B) \(\frac{1}{r}\) (C) \(\frac{1}{r^{2}}\) (D) \(\frac{1}{r^{3}}\)

A swimmer wishes to cross a \(800 \mathrm{~m}\) wide river flowing at \(6 \mathrm{~km} / \mathrm{hr}\). His speed with respect to water is \(4 \mathrm{~km} / \mathrm{hr}\). He crosses the river in shortest possible time. He is drifted downstream on reaching the other bank by a distance of (A) \(800 \mathrm{~m}\) (B) \(1200 \mathrm{~m}\) (C) \(400 \sqrt{13} \mathrm{~m}\) (D) \(2000 \mathrm{~m}\)

A projectile is thrown horizontally from top of a building of height \(10 \mathrm{~m}\) with certain speed \((u)\). At the same time another projectile is thrown from ground \(10 \mathrm{~m}\) away from the building with equal speed \((u)\) on the same vertical plane. If they collide after \(2 s\), then choose the correct options. (A) The angle of projection for second projectile is \(60^{\circ}\) and \(u=10 \mathrm{~ms}^{-1}\) (B) The angle of projection for second projectile is \(90^{\circ}\) and \(u=5 \mathrm{~ms}^{-1}\) (C) The angle of projection for second projectile is \(60^{\circ}\) and \(u=5 \mathrm{~ms}^{-1}\) (D) The angle of projection for second projectile is \(45^{\circ}\) and \(u=10 \mathrm{~ms}^{-1}\)
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