Question
The displacement of an elastic wave is given by the function $\text{y}=3\sin\omega\text{t}+4\cos\omega\text{t}$ where y is in cm and t is in second. Calculate the resultant amplitude.
✓
Answer
$\because\text{y}=3\sin\omega\text{t}+4\cos\omega\text{t}\ ...(\text{i})$Let $3=\text{a}\cos\phi\ ...(\text{ii})$
$4=\text{a}\sin\phi\ ...{\text{iii}}$
Then $\text{y}=\text{a}\cos\phi\sin\omega\text{t}+\text{a}\sin\phi\cos\omega\text{t}$
$\text{y}=\text{a}\sin(\omega\text{t}+\phi) $
From (ii) and(iii)
$\tan \phi=\frac{4}{3}$ or $\phi=\tan^{-1}\frac{4}{3}$
On squaring and adding (ii) and (iii) equations
$\text{a}_2\cos^2\phi+\text{a}^2\sin^2\phi=3^2+4^2$
$\text{a}^2(\cos^2\phi+\sin^2\phi)=9+16$
$\text{a}^2=25\Rightarrow\text{a}=5$
$\text{y}'=5\sin(\omega\text{t}+\phi)$ when $\phi=\tan^{-1}\frac{4}{3}$
Hence, New amplitude is 5 cm.
$4=\text{a}\sin\phi\ ...{\text{iii}}$
Then $\text{y}=\text{a}\cos\phi\sin\omega\text{t}+\text{a}\sin\phi\cos\omega\text{t}$
$\text{y}=\text{a}\sin(\omega\text{t}+\phi) $
From (ii) and(iii)
$\tan \phi=\frac{4}{3}$ or $\phi=\tan^{-1}\frac{4}{3}$
On squaring and adding (ii) and (iii) equations
$\text{a}_2\cos^2\phi+\text{a}^2\sin^2\phi=3^2+4^2$
$\text{a}^2(\cos^2\phi+\sin^2\phi)=9+16$
$\text{a}^2=25\Rightarrow\text{a}=5$
$\text{y}'=5\sin(\omega\text{t}+\phi)$ when $\phi=\tan^{-1}\frac{4}{3}$
Hence, New amplitude is 5 cm.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The pulleys in figure are identical, each having a radius $R$ and moment of inertia $I$. Find the acceleration of the block $M$.
→Draw the current-voltage characteristics for the device shown in figure. between the terminals A and B.
A man runs across the roof-top of a tall building and jumps horizontally with the hope of landing on the roof of the next building which is of a lower height than the first. If his speed is $9m/s$, the (horizontal) distance between the two buildings is 10m and the height difference is $9m$, will he be able to land on the next building? (take $g = 10m/s^2$).
→Explain the concept of potential energy.
A block of mass 1kg is pushed up a surface inclined to horizontal at an angle of 30º by a force of 10N parallel to the inclined surface. The coefficient of friction between block and the incline is 0.1. If the block is pushed up by 10m along the incline, calulate.
- Work done against gravity.
- Work done against force of friction.
- Increase in potential energy.
- Increase in kinetic energy.
- Work done by applied force.
Find the rate of heat flow through a cross section of the rod shown in figure, $\big(\theta_2>\theta_1\big).$ Thermal conductivity of the material of the rod is K.
→A hammer of mass $M$ drops from a height $h.$ It strikes a nail placed vertically on the ground and drives it into the ground through a distance $D$. Calculate.
→- The average resistance offered by the ground, assuming that the hammer and nail remain stuck together after impact.
- The time for which the nail is in motion and.
- The loss in kinetic energy in impact.
A pipe $20cm$ long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a $430Hz$ source? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is $340m s^{–1}$).
→The factors affecting the time period of a simple pendulum are mass, length and the acceleration due to gravity. Deduce a relation for the time period of a simple pendulum.
Let $\text{X}=\overline{\text{ABC}}+\overline{\text{BCA}}+\overline{\text{CAB}}.$ Evaluate X for:
→- $\text{A}=1,\text{B}=0,\text{C}=1$
- $\text{A}=\text{B}=\text{C}=1$
- $\text{A}=\text{B}=\text{C}=0$