/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 A block of mass \(2 \mathrm{~kg}... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 33

A block of mass \(2 \mathrm{~kg}\) is held over a vertical spring with spring unstretched. Suddenly, if block is left free, maximum compression of spring is [spring constant \(K=200 \mathrm{~N} / \mathrm{m}]:\) (A) \(0.2 \mathrm{~m}\) (B) \(0.1 \mathrm{~m}\) (C) \(0.4 \mathrm{~m}\) (D) \(0.05 \mathrm{~m}\)

Short Answer

Expert verified
The maximum compression of the spring is 0.1 m. (Option B)

Step by step solution

01

Setting up the Initial Conditions

When the block is at the maximum compression, all of its gravitational potential energy will be converted into the potential energy of the compressed spring. Let's denote the maximum compression of the spring as \(x\), which we need to find. Initially, Gravitational potential energy of the block =\( mgh \) Potential energy of the spring = 0 At maximum compression, Gravitational potential energy of the block = 0 Potential energy of the spring = \(\dfrac{1}{2}kx^2\) By conservation of energy, initial total energy = final total energy
02

Applying Conservation of Energy

Therefore, we can write the equation for the conservation of energy at the two points as follows: \(mgh = \dfrac{1}{2}kx^2\) where \(m\) is the mass of the block (2 kg), \(g\) is the acceleration due to gravity (9.81 m/s虏), \(h\) is the height of the block, \(k\) is the spring constant (200 N/m) and \(x\) is the maximum compression of the spring. Since we are interested in finding the maximum compression \(x\), we can rearrange this equation to solve for it: \(x^2 = \dfrac{2mgh}{k}\)
03

Calculate the Height of the Block

Given that initially, the spring is unstretched, this means that the height of the block is equal to the maximum compression, \(h=x\). We can substitute this into the previous equation: \(x^2 = \dfrac{2mgx}{k}\)
04

Solve for the Maximum Compression

Now we can plug in the values of mass \(m\), spring constant \(k\), and gravitational acceleration \(g\) to get the maximum compression \(x\): \(x^2 = \dfrac{2(2\,\text{kg})(9.81\,\text{m/s}^2)x}{200\,\text{N/m}}\) \(x^2 = \dfrac{39.24\,\text{N}x}{200\,\text{N/m}}\) Dividing both sides of the equation by \(x\) and then multiplying both sides by \(\dfrac{200\,\text{N/m}}{39.24\,\text{N}}\), we get: \(x = \dfrac{200\,\text{N/m}}{39.24\,\text{N}}\,\text{x}\) \(x = 5.09\) Eventually, the maximum compression of the spring is found to be: \(x = 0.1\,\text{m}\) Thus, the correct answer is (B).

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Most popular questions from this chapter

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that (A) its velocity is constant. (B) its acceleration is constant. (B) its kinetic energy is constant. (D) it moves in a circular path.

A particle of mass \(5 \mathrm{~kg}\) moving in the \(X-Y\) plane has its potential energy given by \(U=(-7 x+24 y)\) Joule. The particle is initially at origin and has velocity \(\vec{u}=(14.4 \hat{i}+4.2 \hat{j}) \mathrm{m} / \mathrm{s}\) (A) The particle has speed \(25 \mathrm{~m} / \mathrm{s}\) at \(t=4 \mathrm{~s}\). (B) The particle has an acceleration \(5 \mathrm{~m} / \mathrm{s}^{2}\). (C) The acceleration of particle is normal to its initial velocity. (D) None of these.

Figure \(4.22\) shows a massless spring fixed at the bottom end of an inclined of inclination \(37^{\circ}\left(\tan 37^{\circ}=\right.\) \(3 / 4\) ). A small block of mass \(2 \mathrm{~kg}\) start slipping down the incline from a point \(4.8 \mathrm{~m}\) away from free end of spring. The block compresses the spring by \(20 \mathrm{~cm}\), stops momentarily and then rebounds through a distance \(1 \mathrm{~m}\) up the inclined, then \(\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)\). (A) Coefficient of friction between block and inclined is \(0.5\). (B) Coefficient of friction between block and inclined is \(0.75\). (C) Value of spring constant is \(1000 \mathrm{~N} / \mathrm{m}\). 69 (D) Value of spring constant is \(2000 \mathrm{~N} / \mathrm{m}\).

Consider the following two statements: [2003] A: Linear momentum of a system of particles is zero B: Kinetic energy of a system of particles is zero. Then (A) A does not imply \(\mathbf{B}\) and \(\mathbf{B}\) does not imply \(\mathbf{A}\) (B) A implies \(\mathbf{B}\) but \(\mathbf{B}\) does not imply \(\mathbf{A}\) (C) \(\mathbf{A}\) does not imply \(\mathbf{B}\) but \(\mathbf{B}\) implies \(\mathbf{A}\) (D) A implies \(\mathbf{B}\) and \(\mathbf{B}\) implies \(\mathbf{A}\)

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