Question

S A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a frictionless, horizontal track (Fig. P15.75). The force constant of the spring is k, and the equilibrium length is l . Assume all portions of the spring oscillate in phase and the velocity of a segment of the spring of length d x is proportional to the distance x from the fixed end; that is, vx = (x / l) v . Also, notice that the mass of a segment of the spring is dm = (m /l) dx . Find (a) the kinetic energy of the system when the block has a speed v.

Answer 1

The kinetic energy of this block-spring when the block has a speed (v) is given by K.E = 1/2 × (M + m/3)v².

What is kinetic energy?

Kinetic energy can be defined as a form of energy that is possessed by a person due to its motion or change in speed (acceleration).

How to calculate kinetic energy?

Mathematically, kinetic energy can be calculated by using this formula:

K.E = 1/2 × mv²

Where:

• K.E represents the kinetic energy.

• m represents the mass.

• v represents the speed or velocity.

Since the mass of a segment of this spring is dm = (m/l) dx, the kinetic energy for each of its segment would be given by:

dK = 1/2 × (dm)Vx²

This ultimately implies that, the kinetic energy of this block-spring when the block has a speed (v) is given by:

K.E = 1/2 × Mv² + 1/2 × ¹∫₀((x²v²/l²)m/ldx

K.E = 1/2 × (M + m/3)v².

Read more on kinetic energy here: brainly.com/question/15848455

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