Advertisements
Advertisements
Question
One end of a light spring of spring constant k is fixed to a wall and the other end is tied to a block placed on a smooth horizontal surface. In a displacement, the work done by the spring is \[\frac{1}{2}k x^2\] . The possible cases are
(a) at spring was initially compressed by a distance x and was finally in its natural length
(b) it was initially stretched by a distance x and and finally was in its natural length
(c) it was initially in its natural length and finally in a compressed position
(d) it was initially in its natural length and finally in a stretched position.
Solution
(a) at spring was initially compressed by a distance x and was finally in its natural length
(b) it was initially stretched by a distance x and and finally was in its natural length
For an elastic spring, the work done is equal to the negative of the change in its potential energy.
When the spring was initially compressed or stretched by a distance x, its potential energy is given by \[\left( P . E . \right)_i = \frac{1}{2}k x^2\].
When it finally comes to its natural length, its potential energy is given by \[\left( P . E . \right)_f = 0\] .
∴ Work done = \[- \left[ \left( P . E . \right)_f - \left( P . E . \right)_i \right] = - \left[ 0 - \frac{1}{2}k x^2 \right] = \frac{1}{2}k x^2\]
APPEARS IN
RELATED QUESTIONS
The potential energy function for a particle executing linear simple harmonic motion is given by V(x) =`(kx^2)/2`, where k is the force constant of the oscillator. For k = 0.5 N m–1, the graph of V(x) versus x is shown in the figure. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches x = ± 2 m.
When a conservative force does positive work on a body, the potential energy of the body ______.
A body of mass 0.5 kg travels in a straight line with velocity v = `ax^(3/2)` where a = `5 m^(1/2)` s-1. What is the work done by the net force during its displacement from x = 0 to x = 2 m?
Two equal masses are attached to the two ends of a spring of spring constant k. The masses are pulled out symmetrically to stretch the spring by a length x over its natural length. The work done by the spring on each mass is
A block of mass m moving at a speed ν compresses a spring through a distance x before its speed is halved. Find the spring constant of the spring.
Consider the situation shown in the following figure. Initially the spring is unstretched when the system is released from rest. Assuming no friction in the pulley, find the maximum elongation of the spring.
A block of mass m sliding on a smooth horizontal surface with a velocity \[\vec{\nu}\] meets a long horizontal spring fixed at one end and with spring constant k, as shown in following figure following. Find the maximum compression of the spring. Will the velocity of the block be the same as \[\vec{\nu}\] when it comes back to the original position shown?
A small block of mass 100 g is pressed against a horizontal spring fixed at one end to compress the spring through 5 cm (figure following). The spring constant is 100 N/m. When released, the block moves horizontally till it leaves the spring. Where will it hit the ground 2 m below the spring?
If a spring extends by x on loading, then energy stored by the spring is (T is the tension in the spring and k is the spring constant)
The potential energy function for a particle executing linear SHM is given by `V(x) = 1/2 kx^2` where k is the force constant of the oscillator (Figure). For k = 0.5 N/m, the graph of V(x) versus x is shown in the figure. A particle of total energy E turns back when it reaches `x = ±x_m`. If V and K indicate the P.E. and K.E., respectively of the particle at `x = +x_m`, then which of the following is correct?
An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of 50,000 kg is moving with a speed of 36 km h–1 when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If 90% of energy of the wagon is lost due to friction, calculate the spring constant.
A curved surface is shown in figure. The portion BCD is free of friction. There are three spherical balls of identical radii and masses. Balls are released from rest one by one from A which is at a slightly greater height than C.
With the surface AB, ball 1 has large enough friction to cause rolling down without slipping; ball 2 has a small friction and ball 3 has a negligible friction.
- For which balls is total mechanical energy conserved?
- Which ball (s) can reach D?
- For balls which do not reach D, which of the balls can reach back A?
Two identical steel cubes (masses 50 g, side 1 cm) collide head-on face to face with a speed of 10 cm/s each. Find the maximum compression of each. Young’s modulus for steel = Y = 2 × 1011 N/m2.
