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Harmonic motions Questions

Questions

A pendulum has a period $14.4\, {\rm s}$ on earth. This same pendulum is taken to the mars and set into oscillation. Its period on mars is $23.1\, {\rm s}$. what is the surface gravity on mars? 

A mass is oscillating on a spring with a period of ${\rm 4.60\ s}$. At $t =0\, {\rm s}$ the mass has zero speed and is at $x=8.30\,{\rm cm}$. What is its speed at $t=2.50\,{\rm s}$?

The position of an air-track cart that is oscillating on a spring is given by $x=\left(12.4\ {\rm cm}\right)\,{\cos  (6.35\ {{\rm s}}^{{\rm -}{\rm 1}})t\ }$. At what value of $t$ after $t=0$ is the car first located at $x=8.47\, {\rm cm}$?

A simple harmonic oscillator has an amplitude of ${\rm 3.50\ cm}$ and a maximum speed of ${\rm 20.0\ cm/s}$. What is its speed when the displacement is ${\rm 1.75\ cm}$?

A $5.88\ {\rm kg}$ mass is connected to a spring and set into oscillation with a period of $2.86\, {\rm s}$ on a frictionless horizontal surface. Whenever this mass passes through the system's equilibrium position, its velocity has an absolute value of $13.3\, {\rm m/s}$. 
(a) What is the total energy of this system?
(b) What is the spring constant of the spring?
(c) What is the amplitude of this oscillation? 

A $0.025\, {\rm kg}$ block on a horizontal frictionless surface is attached to an ideal massless spring whose spring constant is $150\ {\rm N/m}$. the block is pulled from its equilibrium position at $x=0\, {\rm m}$ to a displacement $x=+0.08\, {\rm m}$ and is released from rest. The block then executes simple harmonic motion (SHO) along the horizontal $x$ axis. When the displacement is $x=0.024\, {\rm m}$, what is the kinetic energy of the bock?

A uniform meter stick is freely pivoted about the $0.2\ {\rm m}$ mark. If it is allowed to swing in a vertical plane with a small amplitude and friction, what is the frequency of its oscillations?
 

 A ${\rm 3.0\ kg}$ mass attached to a spring oscillates on a frictionless table with an amplitude $A\ =\ 8.0\times {10}^{-2}\ {\rm m}$. Its maximum acceleration is ${\rm 3.5\ m/}{{\rm s}}^{{\rm 2}}$ . What is the total energy of the system? 

One block of mass ${\rm 1\ kg}$ is connected on two sides by two springs of different stiffnesses. The block rests on a frictionless surface. The block is displaced by ${\rm 5\ cm}$ from its equilibrium position and released from rest. 


(a) What is the frequency of ensuing oscillation? 
(b) What is the speed of the block when it passes the equilibrium position?
(c) When the block is passing through the equilibrium point where the springs are unstrained, another block of equal mass and speed as the first block is attached to it. What is the new amplitude of the oscillation? 

A steel wire $2.00\, {\rm m}$ long with a circular cross section must stretch no more than $0.2\ {\rm cm}$ when a $900\, {\rm N}$ weight is hung from one of its ends. Calculate the minimum diameter that this wire must have. ( Young's modulus of steel is $2\times {10}^{11}{\rm \ Pa\ }$) 

A harmonic oscillator is made by using a $0.3\, {\rm kg}$ frictionless block and an ideal spring with an unknown force constant. The oscillator is found to have a period of $0.2\, {\rm s}$. 
(a) Calculate the force constant of the spring.
(b) The block is now pulled back, so that its maximum displacement from equilibrium is $0.05\, {\rm m}$ and it is released from this point. Calculate the elastic energy in the spring at this maximum displacement.
(c) Using conservation of energy, calculate the maximum velocity of the block, as it passes through the equilibrium position (where the displacement is zero). 

A cup of mass $m=0.2\, {\rm kg}$ rests on a frictionless horizontal surface and is mounted between two springs of equal constant $k_1=100\, {\rm N/m}$ as shown in the figure below. If the cup of mass $m$ is displaced a distance $A=0.1\, {\rm m}$ and released, it will undergo simple harmonic motion. 
(a) What is the period of its oscillation?
Consider now what happens if a piece of putty of mass $m/2$ is dropped into the cup and sticks to it. If this occurs when the right spring is at its maximum compression. 
(b) What is now the amplitude of oscillation?
(c) What is now the frequency of the oscillation?
If this occurs when the block is moving through its equilibrium position 
(d) How much energy was lost in the collision?
(e) What is now the amplitude of the motion?

This plot shows a mass oscillating as $x=x_m\,\cos \left(\omega t-\phi \right)$. What are $x_m$ and $\phi$
    (a) $1 {\mathrm m}\,.\,0^\circ$ 
    (b) $2 {\mathrm m}\,.\,0^\circ$
    (c) $2 {\mathrm m}\,.\,90^\circ$
    (d) $4 {\mathrm m}\,.\,0^\circ$

All of the relevant quantities in a simple harmonic motion are related together by the following basic equation:
\[x(t)=A\,\sin (\omega t\pm \phi_0) \quad  or \quad x(t)=A\,\cos (\omega t\pm \phi_0)\]
Where $(\omega t\pm \phi_0)$ is called the phase of oscillation. $\omega=\frac {2\pi}{T}=2\pi f$ is the angular frequency of oscillation. $\phi_0$ is the phase constant of the motion which is determined by the initial condition by setting $t=0$ in equation above
\[t=0 \to x\left(t=0\right)=A\,\cos (0\pm \phi_0) \to \phi_0=\cos^{-1} \left(\frac {x(0)}A\right)\]
Note: in graphs, the amplitude is the distance from the axis to the maximum point.
From the graph, the oscillating mass start from $x(t=0)=0$, therefore 
\begin{gather*}
x_m=A=2 {\mathrm m} \\
and \quad \phi_0=\cos^{-1} \left(\frac{x(t=0)}A\right)=\cos^{-1} \left(\frac 02\right)=\cos^{-1}(0)\\
\therefore \ \ \phi_0=\cdots,-270^\circ,-90^\circ,90^\circ,270^\circ,\cdots
\end{gather*}

The correct answer is C. 
 

A particles moves in simple harmonic motion according to $x=2\,{\,\cos (50t)\ }$, where $x$ is in meters and $t$ is in seconds. Its maximum velocity is:
(a) $100\,{\,\sin \left(50t\right)\ }\frac{\mathrm{m}}{\mathrm{s}}$
(b) $100\,{\,\cos \left(50t\right)\ }\frac{\mathrm{m}}{\mathrm{s}}$
(b) $100\frac{\mathrm{m}}{\mathrm{s}}$
(c) $200\frac{\mathrm{m}}{\mathrm{s}}$
(d) None of these

If the length of a simple pendulum is doubled, its period will:
(a) Halve
(b) Increase by a factor of $\sqrt{2}$
(c) Decrease by a factor of $\sqrt{2}$
(d) Double
(e) Remain the same
 


Harmonic motions
Category : Harmonic motions

MOST USEFUL FORMULA IN HARMONIC MOTIONS:

Period $T$ is time for one cycle.
Frequency is the number of cycles per unit time
\[f=\frac 1 T\]
Angular frequency $\omega=2\pi f=\frac {2\pi}{T}$

Restoring force in Simple Harmonic Motions(SHM):
\[F_x=-kx\]
Angular frequency in SHM:
\[\omega=\sqrt{\frac k m}\]
Displacement in SHM: 
\[x(t)=A\,\cos \left(\omega t \pm \phi\right)\]
Velocity in SHM:
\[v=\frac{dx}{dt}=-A\omega\,\cos \left(\omega t \pm \phi\right)\]
Acceleration in SHM:
\[a=\frac {d^{2}x}{dt^{2}}=A\omega^2\,\cos \left(\omega t \pm \phi\right)\]
where $A$ is the maximum distance from equilibrium (amplitude)

Energy in SHM:
\[E=\frac 1 2 mv_x^2+\frac 1 2 kx^2=\frac 1 2 kA^2\]
Angular frequency in simple pendulum:
\[\omega=\sqrt{\frac g L}\]
Frequency of a simple pendulum:
\[f=\frac \omega {2\pi}=\frac 1{2\pi} \sqrt{\frac g L}\]
Angular frequency in physical pendulum:
\[\omega=\sqrt{\frac {mgd}{I}}\]
where $d$ and $I$ are the distance from axis of rotation to the center of gravity and moment of inertia about the axis,respectively. 


Number Of Questions : 15