Advertisements
Advertisements
Question
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.
Advertisements
Solution
As the block of mass M is pulled, a net resultant force is exerted by the three springs opposing the motion of the block.
Now, springs k2 and k3 are in connected as a series combination.
Let k4 be the equivalent spring constant.
\[\therefore \frac{1}{k_4} = \frac{1}{k_2} + \frac{1}{k_3} = \frac{k_2 + k_3}{k_2 k_3}\]
\[ k_4 = \frac{k_2 k_3}{k_2 + k_3}\]
k4 and k1 form a parallel combination of springs. Hence, equivalent spring constant k = k1 + k4.
\[= \frac{k_2 k_3}{k_2 + k_3} + k_1 \]
\[ = \frac{k_2 k_3 + k_1 k_2 + k_1 k_3}{k_2 + k_3}\]
\[ \therefore \text { Time peiod }, T = 2\pi\sqrt{\frac{M}{k}}\]
\[ = 2\pi\sqrt{\frac{M \left( k_2 + k_3 \right)}{k_2 k_3 + k_1 k_2 + k_1 k_3}}\]
(b) Frequency \[\left( v \right)\] is given by,
\[v = \frac{1}{T}\]
\[= \frac{1}{2\pi}\sqrt{\frac{k_2 k_3 + k_1 k_2 + k_1 k_3}{M\left( k_2 + k_3 \right)}}\]
(c) Amplitude ( x ) is given by,
\[x = \frac{F}{k} = \frac{F\left( k_2 + k_3 \right)}{k_1 k_2 + k_2 k_3 + k_1 k_3}\]
APPEARS IN
RELATED QUESTIONS
A particle having mass 10 g oscillates according to the equation x = (2.0 cm) sin [(100 s−1)t + π/6]. Find (a) the amplitude, the time period and the spring constant. (c) the position, the velocity and the acceleration at t = 0.
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 ?
A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum, i.e., a pendulum having frequency same as that of the block.
Solve the previous problem if the pulley has a moment of inertia I about its axis and the string does not slip over it.
A 1 kg block is executing simple harmonic motion of amplitude 0.1 m on a smooth horizontal surface under the restoring force of a spring of spring constant 100 N/m. A block of mass 3 kg is gently placed on it at the instant it passes through the mean position. Assuming that the two blocks move together, find the frequency and the amplitude of the motion.
Find the elastic potential energy stored in each spring shown in figure when the block is in equilibrium. Also find the time period of vertical oscillation of the block.
Discuss in detail the energy in simple harmonic motion.
Show that for a particle executing simple harmonic motion.
- the average value of kinetic energy is equal to the average value of potential energy.
- average potential energy = average kinetic energy = `1/2` (total energy)
Hint: average kinetic energy = <kinetic energy> = `1/"T" int_0^"T" ("Kinetic energy") "dt"` and
average potential energy = <potential energy> = `1/"T" int_0^"T" ("Potential energy") "dt"`
When a particle executing S.H.M oscillates with a frequency v, then the kinetic energy of the particle?
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.
A body of mass m is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is 4 cm below the point, where it was held in hand.
What is the amplitude of oscillation?
A particle undergoing simple harmonic motion has time dependent displacement given by x(t) = A sin`(pit)/90`. The ratio of kinetic to the potential energy of this particle at t = 210s will be ______.
