Advertisements
Advertisements
प्रश्न
Repeat the previous exercise if the angle between each pair of springs is 120° initially.
Advertisements
उत्तर
As the particle is pushed against the spring C by the distance x, it experiences a force of magnitude kx.
If the angle between each pair of the springs is 120˚ then the net force applied by the springs A and B is given as,
\[\sqrt{\left( \frac{kx}{2} \right)^2 + \left( \frac{kx}{2} \right)^2 + 2\left( \frac{kx}{2} \right)\left( \frac{kx}{2} \right) \cos 120^\circ} = \frac{kx}{2}\]
Total resultant force \[\left( F \right)\] acting on mass m will be,
\[F = kx + \frac{kx}{2} = \frac{3kx}{2} \]
\[ \therefore a = \frac{F}{m} = \frac{3kx}{2m}\]
\[ \Rightarrow \frac{a}{x} = \frac{3k}{2m} = \omega^2 \]
\[ \Rightarrow \omega = \sqrt{\frac{3k}{2m}}\]
\[ \therefore \text { Time period }, T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{2m}{3k}}\]
APPEARS IN
संबंधित प्रश्न
A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is
(a) at the end A,
(b) at the end B,
(c) at the mid-point of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.
The maximum speed and acceleration of a particle executing simple harmonic motion are 10 cm/s and 50 cm/s2. Find the position(s) of the particle when the speed is 8 cm/s.
Consider a particle moving in simple harmonic motion according to the equation x = 2.0 cos (50 πt + tan−1 0.75) where x is in centimetre and t in second. The motion is started at t = 0. (a) When does the particle come to rest for the first time? (b) When does he acceleration have its maximum magnitude for the first time? (c) When does the particle come to rest for the second time ?
The pendulum of a clock is replaced by a spring-mass system with the spring having spring constant 0.1 N/m. What mass should be attached to the spring?
The block of mass m1 shown in figure is fastened to the spring and the block of mass m2 is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The blocks are pushed a further distance (2/k) (m1 + m2)g sin θ against the spring and released. Find the position where the two blocks separate. (c) What is the common speed of blocks at the time of separation?
The spring shown in figure is unstretched when a man starts pulling on the cord. The mass of the block is M. If the man exerts a constant force F, find (a) the amplitude and the time period of the motion of the block, (b) the energy stored in the spring when the block passes through the equilibrium position and (c) the kinetic energy of the block at this position.
The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block.
Consider the situation shown in figure . Show that if the blocks are displaced slightly in opposite direction and released, they will execute simple harmonic motion. Calculate the time period.
When a particle executing S.H.M oscillates with a frequency v, then the kinetic energy of the particle?
If a body is executing simple harmonic motion and its current displacements is `sqrt3/2` times the amplitude from its mean position, then the ratio between potential energy and kinetic energy is:
A body is executing simple harmonic motion with frequency ‘n’, the frequency of its potential energy is ______.
A body is executing simple harmonic motion with frequency ‘n’, the frequency of its potential energy is ______.
A body is executing simple harmonic motion with frequency ‘n’, the frequency of its potential energy is ______.
A body is performing S.H.M. Then its ______.
- average total energy per cycle is equal to its maximum kinetic energy.
- average kinetic energy per cycle is equal to half of its maximum kinetic energy.
- mean velocity over a complete cycle is equal to `2/π` times of its π maximum velocity.
- root mean square velocity is times of its maximum velocity `1/sqrt(2)`.
Displacement versus time curve for a particle executing S.H.M. is shown in figure. Identify the points marked at which (i) velocity of the oscillator is zero, (ii) speed of the oscillator is maximum.
Find the displacement of a simple harmonic oscillator at which its P.E. is half of the maximum energy of the oscillator.
A mass of 2 kg is attached to the spring of spring constant 50 Nm–1. The block is pulled to a distance of 5 cm from its equilibrium position at x = 0 on a horizontal frictionless surface from rest at t = 0. Write the expression for its displacement at anytime t.
The total energy of a particle, executing simple harmonic motion is ______.
where x is the displacement from the mean position, hence total energy is independent of x.
