Advertisements
Advertisements
Question
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?
Advertisements
Solution
(a) As it can be seen from the figure,
Restoring force = kx
Component of total weight of the two bodies acting vertically downwards = (m1 + m2) g sin θ
At equilibrium,
kx = (m1 + m2) g sin θ
\[\Rightarrow x = \frac{\left( m_1 + m_2 \right) g \sin \theta}{k}\]
(b) It is given that:
Distance at which the spring is pushed,
\[x_1 = \frac{2}{k}\left( m_1 + m_2 \right)g \sin \theta\]
As the system is released, it executes S.H.M.
where \[\omega = \sqrt{\frac{k}{m_1 + m_2}}\]
When the blocks lose contact, P becomes zero. (P is the force exerted by mass m1 on mass m2)
\[\therefore m_2 g \sin \theta = m_2 x_2 \omega^2 = m_2 x_2 \times \frac{k}{m_1 + m_2}\] \[ \Rightarrow x_2 = \frac{\left( m_1 + m_2 \right) g \sin \theta}{k}\]
Therefore, the blocks lose contact with each other when the spring attains its natural length.
(c) Let v be the common speed attained by both the blocks.
\[\text { Total compression } = x_1 + x_2 \]
\[\frac{1}{2} \left( m_1 + m_2 \right) v^2 - 0 = \frac{1}{2}k \left( x_1 + x_2 \right)^2 - \left( m_1 + m_2 \right) g \sin \theta \left( x + x_1 \right) \]
\[ \Rightarrow \frac{1}{2}\left( m_1 + m_2 \right) v^2 = \frac{1}{2}k\left( \frac{3}{k} \right) \left( m_1 + m_2 \right) g \sin \theta - \left( m_1 + m_2 \right) g \sin \theta \left( x_1 + x_2 \right)\]
\[ \Rightarrow \frac{1}{2}\left( m_1 + m_2 \right) v^2 = \frac{1}{2} \left( m_1 + m_2 \right) g \sin \theta \times \left( \frac{3}{k} \right) \left( m_1 + m_2 \right) g sin \theta\]
\[ \Rightarrow v = \sqrt{\left\{ \frac{3}{k} \left( m_1 + m_2 \right) \right\}}g \sin \theta\]
APPEARS IN
RELATED QUESTIONS
A particle executes simple harmonic motion with an amplitude of 10 cm. At what distance from the mean position are the kinetic and potential energies equal?
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.
The equation of motion of a particle started at t = 0 is given by x = 5 sin (20t + π/3), where x is in centimetre and t in second. When does the particle
(a) first come to rest
(b) first have zero acceleration
(c) first have maximum speed?
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.
A body of mass 2 kg suspended through a vertical spring executes simple harmonic motion of period 4 s. If the oscillations are stopped and the body hangs in equilibrium find the potential energy stored in the spring.
In following figure k = 100 N/m M = 1 kg and F = 10 N.
- Find the compression of the spring in the equilibrium position.
- A sharp blow by some external agent imparts a speed of 2 m/s to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant.
- Find the time period of the resulting simple harmonic motion.
- Find the amplitude.
- Write the potential energy of the spring when the block is at the left extreme.
- Write the potential energy of the spring when the block is at the right extreme.
The answer of b, e and f are different. Explain why this does not violate the principle of conservation of energy.
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.
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 rectangle plate of sides a and b is suspended from a ceiling by two parallel string of length L each in Figure . The separation between the string is d. The plate is displaced slightly in its plane keeping the strings tight. Show that it will execute simple harmonic motion. Find the time period.
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.
When the displacement of a particle executing simple harmonic motion is half its amplitude, the ratio of its kinetic energy to potential energy is ______.
A body is executing simple harmonic motion with frequency ‘n’, the frequency of its potential energy is ______.
Motion of an oscillating liquid column in a U-tube is ______.
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.
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?
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.
