Karnataka Board PUCPUC Science Class 11

A Uniform Plate of Mass M Stays Horizontally and Symmetrically on Two Wheels Rotating in Opposite Direction in Figure . - Physics

Advertisements
Advertisements
Sum

A uniform plate of mass M stays horizontally and symmetrically on two wheels rotating in opposite direction in Figure . The separation between the wheels is L. The friction coefficient between each wheel and the plate is μ. Find the time period of oscillation of the plate if it is slightly displaced along its length and released.

Advertisements

Solution

Let x be the displacement of the uniform plate towards left.
Therefore, the centre of gravity will also be displaced through displacement x.
At the displaced position,
R1 + R2 = mg
Taking moment about g, we get:

\[R_1 \left( \frac{L}{2} - x \right) =  R_2 \left( \frac{L}{2} + x \right) = \left( Mg - R_1 \right)  \left( \frac{L}{2} + x \right)               .  .  .  . \left( 1 \right)\] 

\[ \therefore    R_1 \left( \frac{L}{2} - x \right) = \left( Mg - R_1 \right)  \left( \frac{L}{2} + x \right)\] 

\[ \Rightarrow  R_1 \frac{L}{2} -  R_1 x = Mg\frac{L}{2} -  R_1 x + Mgx -  R_1 \frac{L}{2}\] 

\[ \Rightarrow  R_1 \frac{L}{2} +  R_1 \frac{L}{2} = Mg\left( x + \frac{L}{2} \right)\] 

\[ \Rightarrow  R_1 \left( \frac{L}{2} + \frac{L}{2} \right) = Mg\left( \frac{2x + L}{2} \right)\] 

\[ \Rightarrow  R_1 L = Mg\left( 2x + L \right)\] 

\[ \Rightarrow  R_1  = Mg\frac{\left( L + 2x \right)}{2L}\] 

\[\text { Now }, \] 

\[   F_1  = \mu R_1  = \mu Mg\frac{\left( L + 2x \right)}{2L}\] 

\[\text { Similarly },   \] 

\[ F_2  = \mu R_2  = \mu Mg\frac{\left( L - 2x \right)}{2L}\] 

\[\text { As }  F_1  >  F_2 ,   \text { we  can  write: }\] 

\[ F_1  -  F_2  = Ma = \left( 2\mu\frac{Mg}{L} \right)x\] 

\[      \frac{a}{x} = 2\mu\frac{g}{L} =  \omega^2 \] 

\[ \Rightarrow \omega = \sqrt{\frac{2\mu g}{L}}\] 

\[\text { Time  period }\left( T \right)\text{  is  given  by }, \] 

\[T = 2\pi\sqrt{\frac{L}{2\mu g}}\]

  Is there an error in this question or solution?
Chapter 12: Simple Harmonics Motion - Exercise [Page 254]

APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 12 Simple Harmonics Motion
Exercise | Q 31 | Page 254

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

A seconds pendulum is suspended in an elevator moving with constant speed in downward direction. The periodic time (T) of that pendulum is _______.


The periodic time of a linear harmonic oscillator is 2π second, with maximum displacement of 1 cm. If the particle starts from extreme position, find the displacement of the particle after π/3  seconds.


A copper metal cube has each side of length 1 m. The bottom edge of the cube is fixed and tangential force 4.2x108 N is applied to a top surface. Calculate the lateral displacement of the top surface if modulus of rigidity of copper is 14x1010 N/m2.


Which of the following example represent periodic motion?

An arrow released from a bow.


Which of the following example represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

General vibrations of a polyatomic molecule about its equilibrium position.


Figure depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?


The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed?


Answer in brief:

Derive an expression for the period of motion of a simple pendulum. On which factors does it depend?


The length of the second’s pendulum in a clock is increased to 4 times its initial length. Calculate the number of oscillations completed by the new pendulum in one minute.


A person goes to bed at sharp 10.00 pm every day. Is it an example of periodic motion? If yes, what is the time period? If no, why?


The total mechanical energy of a spring-mass system in simple harmonic motion is \[E = \frac{1}{2}m \omega^2 A^2 .\] Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude A remains the same. The new mechanical energy will


A particle executes simple harmonic motion with a frequency v. The frequency with which the kinetic energy oscillates is


Two bodies A and B of equal mass are suspended from two separate massless springs of spring constant k1 and k2 respectively. If the bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of A to that of B is


A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is


The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitude 2 cm, 1 m s−1 and 10 m s−2 at a certain instant. Find the amplitude and the time period of the motion.


A small block of mass m is kept on a bigger block of mass M which is attached to a vertical spring of spring constant k as shown in the figure. The system oscillates vertically. (a) Find the resultant force on the smaller block when it is displaced through a distance x above its equilibrium position. (b) Find the normal force on the smaller block at this position. When is this force smallest in magnitude? (c) What can be the maximum amplitude with which the two blocks may oscillate together?


The left block in figure moves at a speed v towards the right block placed in equilibrium. All collisions to take place are elastic and the surfaces are frictionless. Show that the motions of the two blocks are periodic. Find the time period of these periodic motions. Neglect the widths of the blocks.


The ear-ring of a lady shown in figure has a 3 cm long light suspension wire. (a) Find the time period of small oscillations if the lady is standing on the ground. (b) The lady now sits in a merry-go-round moving at 4 m/s1 in a circle of radius 2 m. Find the time period of small oscillations of the ear-ring.


Find the time period of small oscillations of the following systems. (a) A metre stick suspended through the 20 cm mark. (b) A ring of mass m and radius r suspended through a point on its periphery. (c) A uniform square plate of edge a suspended through a corner. (d) A uniform disc of mass m and radius r suspended through a point r/2 away from the centre.


A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period?


A body of mass 1 kg is mafe to oscillate on a spring of force constant 16 N/m. Calculate (a) Angular frequency, (b) Frequency of vibrations.


The period of oscillation of a body of mass m1 suspended from a light spring is T. When a body of mass m2 is tied to the first body and the system is made to oscillate, the period is 2T. Compare the masses m1 and m2


A 20 cm wide thin circular disc of mass 200 g is suspended to rigid support from a thin metallic string. By holding the rim of the disc, the string is twisted through 60° and released. It now performs angular oscillations of period 1 second. Calculate the maximum restoring torque generated in the string under undamped conditions. (π3 ≈ 31)


Find the number of oscillations performed per minute by a magnet is vibrating in the plane of a uniform field of 1.6 × 10-5 Wb/m2. The magnet has a moment of inertia 3 × 10-6 kg/m2 and magnetic moment 3 A m2.


The maximum speed of a particle executing S.H.M. is 10 m/s and maximum acceleration is 31.4 m/s2. Its periodic time is ______ 


A simple pendulum is inside a spacecraft. What will be its periodic time? 


Which of the following example represent periodic motion?

A hydrogen molecule rotating about its center of mass.


Which of the following example represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

A motion of an oscillating mercury column in a U-tube.


When two displacements represented by y1 = a sin(ωt) and y2 = b cos(ωt) are superimposed the motion is ______. 


A simple pendulum of frequency n falls freely under gravity from a certain height from the ground level. Its frequency of oscillation.


What are the two basic characteristics of a simple harmonic motion?


The time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration `g/2`, the time period of the pendulum will be ______.


When a particle executes Simple Harmonic Motion, the nature of the graph of velocity as a function of displacement will be ______.


A particle performs simple harmonic motion with a period of 2 seconds. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is `1/a` s. The value of 'a' to the nearest integer is ______.


Share
Notifications



      Forgot password?
Use app×