4.100 Maximum acceleration in simple harmonic motion

4.100: A mass of $4\: kg$ suspended from a spring of force constant $800 \:Nm^{-1}$ executes simple harmonic oscillations.   If the total energy of the oscillator is $4\:J$, determine the maximum acceleration in $ ms^{-2}$  of the mass. Solution: Angular frequency of...

4.99 Maximum speed in simple harmonic motion

 A spring of force constant $1200 \:N m^{-1}$ is mounted on a horizontal table as shown in figure. A mass of $3 \:kg$ is attached to its free end and pulled sideways to a distance of $2\: cm$ and released. Calculate the maximum velocity of the mass.   Solution: When...

4.98 Length of a pendulum

Problem 4.98: A simple pendulum is released at rest while its string is making an angle of $10^o$ with the vertical. $0.2$ seconds later the string makes an angle of $5^o$ with the vertical. Determine the length of the pendulum. Take acceleration due to gravity as...

4.97 Time period of oscillation

Problem 4.97: A block of mass $M$ kept on a smooth horizontal surface is connected to three identical springs each with constant $k$. Determine the time period for small amplitude oscillations. Neglect all dissipating forces. Solution: When the block is displaced to...

4.96 Amplitude of oscillation of a spring

Problem 4.96: A block of mass $M$ is connected to a series connection of two springs with spring constants $k_1$ and $k_2$ respectively. This is shown in the diagram. If the block oscillates in a simple harmonic fashion with amplitude determine the amplitude of...

4.94 Amplitude of oscillation

Problem 4.94: A block of mass $M$ on a smooth horizontal surface is connected to a mass less spring of length $L$ and spring constant $k$. The block is displaced by a distance $A$ and released. What is the amplitude of oscillation of the midpoint $P$ of the spring?...

4.93 Total energy of a particle in SHM

Problem 4.93: A block of mass $0.2\:kg$ is attached to a vertical spring of constant $100\:Nm^{-1}$. When the block is at rest it is at a height of $2\:m$ from the ground below. At time $t\:=\:0$ the particle starts simple harmonic motion given by the equation...

4.92 Change in momentum in simple harmonic motion

Problem 4.92: A block of mass $M$ connected to a spring of constant $K$ is resting on a smooth horizontal surface ass shown. The block is pulled from this position through a distance $X$ and then released. Determine the change in momentum as the block crosses the...

4.89 Amplitude of simple harmonic motion

Problem 4.89: The displacement of a particle varies according to the relation $x\: =\: 4 (cos \pi t + sin \pi t)$.  Determine the amplitude of the motion of the particle. Solution: The equation represents the sum of two simple harmonic motions. One of it is $x\: =\: 4...

4.88 Force in simple harmonic motion

Problem 4.88: A block of mass $15\: kg$ executes SHM under the restoring force of a spring. The amplitude and the time period of the motion are $0.1 \:m$ and $3.14 \:s$ respectively. Find the maximum force exerted by the spring on the block. Solution: In simple...

4.87 Length of seconds pendulum

Problem 4.87: What is the length of a simple pendulum, which ticks seconds ? Take acceleration due to gravity as $9.8\:ms^{-2}$. Solution: Time period of a simple pendulum with length $L$ is given by, $T\:=\:2\pi \sqrt{\frac{L}{g}}$. When a pendulum ticks second, each...

4.87 Beat of sound waves

Problem 4.87: Two sound waves having wavelength of $50\:cm$ and $51\:cm$ produces $12$ beats per second. Determine the speed sound in the medium. Solution: Wavelength $\lambda$, frequency $\nu$ and speed $v$ of a wave are related as $v\:=\:\nu\:\lambda$. So frequency...

4.86 Simple harmonic motion

Problem 4.86: A block of mass $M\:kg$ is attached to a mass-less spring of force constant $K$ on a smooth surface as shown in the diagram. A particle of mass $M\:kg$ is projected from the block with a speed of $10\:ms^{_1}$ at an angle of $60^0$ with the horizontal....

4.85 Change in linear momentum in SHM

Problem 4.85: A particle of mass $M$ is in simple harmonic motion of amplitude $A$ and time period $T$ along a straight line. Determine the change in momentum when the particle goes from the mean position to one of the extremes.  Solution: In the case of simple...

4.84 Fundamental frequency of a string

Problem 4.84: A $100 \:cm$ long wire of mass $40\:g$  supports a block of mass of $1.6 \:kg$ as shown. Find the fundamental frequency of the portion of the string between the wall and the pulley.  Take $g \:=\: 10\:ms^{-1}$.   Solution: Fundamental frequency of a...