Question

If particle move under the effect of superposition of two SHM simultaneously and it is given \(x=10(1-\cos \pi t) m\) and \(y=10 \sin \pi t \mathrm{~m}:(\pi=22 / 7)\) then distance travelled by a particle in \(\frac{7}{22} \mathrm{sec}\). (1) \(10 \mathrm{~m}\) (2) \(20 \mathrm{~m}\) (3) \(70 \mathrm{~m}\) (4) zero

Short answer

The distance travelled by the particle is 10 meters.

Step-by-step solution

Analyze given functions

The displacement in the x-direction is given by \(x=10(1-\cos( \pi t) )\ m\), and the displacement in the y-direction is given by \(y=10 \sin( \pi t ) \ m\).

Calculate x and y displacements at given time

To find the displacements at \(t=\frac{7}{22} \ \mathrm{sec}\):Substitute \(t= \frac{7}{22}\) into the x and y displacement functions:\(x=10(1- \cos( \- \pi \frac{7}{22} )\ m\)\(y=10 \sin( \- \pi \frac{7}{22} )\ m\)

Evaluate trigonometric functions

Using the given value of \(\pi = \frac{22}{7}\), calculate the argument for the trigonometric functions:\(\pi \cdot \frac{7}{22} = 1\)Thus, substitute back into the displacement equations:\(\cos(1)\) and \(\sin( 1 )\)

Compute numeric values

Now calculate: \(\cos ( 1 ) = 0.5403\) and \(\sin ( 1 ) = 0.8415 \)Thus:\( x = 10(1 - 0.5403) = 4.597 m\)\( y = 10*0.8415 = 8.4149 m\)

Find displacement vector

The total displacement vector is given by:\[ d^2 = x^2 + y^2 \] Thus, substituting the computed displacements:\[ d = \sqrt{4.597^2 + 8.4149^2} \approx 10 m \]

Distance travelled

The distance travelled by the particle necessarily must be the magnitude of the displacement vector. Hence, the distance travelled by the particle within the given time is 10 meters.

Key concepts

Trigonometric Functions and their Role in SHM

In Simple Harmonic Motion (SHM), trigonometric functions play a crucial role. In our problem, the particle's oscillations in the x and y directions are described using the cosine and sine functions, respectively.
When dealing with SHM, it's common to use sine and cosine because they naturally describe oscillatory motion.

• The cosine function, \(\text{cos}( \pi t)\), represents the x-direction displacement. It shows how the position varies periodically over time.
• The sine function, \(\text{sin}( \pi t)\), describes the y-direction displacement. Like cosine, it varies periodically but starts at zero.

The relationship between these trigonometric functions and time leads to smooth and predictable oscillations. In this problem, substituting specific values for time into these functions helps us find the position of the particle at any given moment.

Understanding Displacement in SHM

Displacement refers to the change in position of the particle from its equilibrium point. It's a vector quantity because it has both magnitude and direction.
In the given exercise, the displacement in the x-direction is given by \(x = 10(1 - \cos( \pi t))\), and in the y-direction by \(y = 10 \sin( \pi t)\).

• At any time \(t\), the expression for \(x\) tells us how far the particle is from its starting point along the x-axis.
• The expression for \(y\) describes how far the particle is along the y-axis.

To find the precise displacement, we substitute the given time value into both functions, resulting in specific x and y positions. These values are crucial for comprehending the particle's overall motion.

Calculating Distance Traveled

Distance traveled is a key concept when analyzing SHM. It tells us exactly how far the particle has moved regardless of direction. It's a scalar quantity, meaning it has only magnitude.
Using the displacements we calculated for both x and y directions at time \(t = \frac{7}{22} \ \text{seconds}\), we find the resultant displacement vector.

Mathematically, the magnitude of this resultant vector is determined by: \[ d = \sqrt{x^2 + y^2} \] Substituting values: \(x = 4.597 \ \text{m}\) and \(y = 8.4149 \ \text{m}\), we get: \[ d = \sqrt{4.597^2 + 8.4149^2} \approx 10 \ \text{m} \] So, the particle travels a total distance of 10 meters in \( \frac{7}{22} \) seconds. Understanding how to sum these distances accurately is vital when handling problems involving SHM.