A block of mass m is attached to two unstretched springs of...

A block of mass $m$ is attached to two unstretched springs of spring constants $k_{1}$ and $k_{2}$ as shown in figure (8-E9). The block is displaced towards right through a distance $x$ and is released. Find the speed of the block as it passes through the mean position shown.

Key Concepts

Conservation of Energy

This principle states that the total energy in a closed system remains constant. In mechanical systems involving springs and masses, the potential energy stored in the deformation of the springs is converted into kinetic energy of the mass, and vice versa. It is a fundamental concept for solving problems where energy is transformed from one form to another without any losses.

Hooke's Law and Spring Potential Energy

Hooke's Law describes the relationship between the force exerted by a spring and its displacement from equilibrium. The potential energy stored in a spring when it is stretched or compressed is given by one-half times the spring constant times the square of the displacement. This concept is crucial in determining how the energy stored in elastic components is released to produce motion.

Effective Spring Constant

When a system involves multiple springs that support a load, an effective spring constant can be defined to describe the overall stiffness of the system. In cases where springs act in parallel, as with springs attached to opposite sides of a mass, the effective spring constant is the sum of the individual spring constants. This allows one to treat the system as a single spring for further analyses, such as calculating energy storage or oscillation frequency.

Transcript

00:01 Again, in this question, a situation is given like this, okay? this is a floor at which a mass is given m, okay? and that is attached with the spring, spring is further attached, further fixed, okay? and right side is again, there is a spring and that is fixed.

00:16 Okay, so the left side, the spring constant is k1 and the right side, the spring constant is k2.

00:23 Understand? and now, what happens? the block is displaced towards right through a distance x okay so when it's and is released okay so when there is a displacement of x here then there is a displacement of x also in the left hand side okay when the mass move right side so we have to find out the block as it passes through the mean position shown okay so it will go first of all there and now when we release this will go again to the left -hand side okay so let's suppose at the mean position the velocity is v okay let's suppose velocity is v so the total energy we can apply the law of conversation of energy conservation of energy okay so the total energy at main point okay that will be the kinetic energy initially okay not initially after the stretching that is half mv square okay half mv square and that is the equals to the energy due to the spring in the first and the second okay so in the first portion that will be half kx square so half k1 x square and in the second portion that is again 1 by 2 k1 k2 x square okay so stretching is the same in the first and second portion okay x and x so from here we can find out the velocity v okay first of all we will eliminate this 1 by 2 in both side of the equation and m v square that is equals to k1 plus k2 x square understand so now v square it will be k1 plus k2 divided by m multiplied by x square okay so now we will do the square root so we will be square root of x square that is x and square root of k1 plus k2 divided by m that can be written as square root of k1 plus k2 divided by m this this will be the final answer of this question...

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