/*! 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 58 A block of mass \(m\) is connect... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 58

A block of mass \(m\) is connected to another block of mass \(M\) by a massless spring of constant \(k\). The blocks are kept on a smooth horizontal plane. Initially, the blocks are at reat and the spring is unstretched. Then a constant force \(F\) starts acting on the block of mass \(M\) to pull it. Find the force on the block of mass \(m\). (a) \(\frac{M F}{m+M}\) (b) \(\frac{m F}{M}\) (c) \(\frac{(M+m) F}{m}\) (d) \(\frac{m F}{m+M}\)

Short Answer

Expert verified
The force on block m is \(\frac{m F}{m+M}\) (Option d).

Step by step solution

01

Analyze System Dynamics

Consider both blocks and the spring as a single system. Since the force \( F \) is applied to block \( M \), the entire system (both masses) will accelerate together. As the spring is massless and only internal forces act between the blocks, the force \( F \) provides the net acceleration.
02

Calculate the System Acceleration

The force \( F \) causes an acceleration of the entire system. Use Newton's second law to find the acceleration:\[ a = \frac{F}{m+M} \]where \( a \) is the acceleration of both blocks together.
03

Determine Force on Block m due to Acceleration

To find the force on the block of mass \( m \), consider its individual motion as it accelerates due to the common acceleration \( a \). Since \( m \) is being dragged along,by Newton's second law on \( m \):\[ F_m = m \cdot a = m \cdot \frac{F}{m+M} \]
04

Match with Given Options

Compare the derived expression \( F_m = \frac{m F}{m+M} \) with the given options. It matches option (d).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Laws of Motion
Newton's Laws of Motion are the foundation for solving problems involving forces and motion. The first law, also known as the law of inertia, tells us that an object at rest will remain at rest unless acted upon by an external force. The second law provides the equation for force, stating that the force on an object is equal to its mass multiplied by its acceleration: \\[ F = m imes a \] \The third law states that for every action, there is an equal and opposite reaction. These laws help us analyze and predict the behavior of objects when forces are applied.
System of Particles and Rigid Body
When dealing with systems of particles, such as the two blocks connected by a spring, it is crucial to consider the system's overall dynamics. Treat the blocks and spring as a unified system enhances the understanding of how forces affect movement. For rigid bodies, each part of the system moves as a whole, maintaining its shape, which simplifies calculations involving internal forces like spring forces. Since the spring is massless, it does not contribute to the system's inertia, allowing us to focus solely on the masses' interaction.
Force and Acceleration
Force and acceleration are deeply interrelated, as given by Newton's second law. In this exercise, a constant force \( F \) is applied to one block, causing the entire system to accelerate. The acceleration can be found using the equation: \\[ a = \frac{F}{m+M} \] \Here, \( m \) and \( M \) are the masses of the blocks, and \( a \) represents their common acceleration. This acceleration must be considered when determining the force exerted on the smaller block, as it moves with this acceleration as a part of the entire system.
Spring Force Dynamics
In spring force dynamics, the force exerted by a spring is related to its extension or compression from the natural length. In a perfect spring, the force exerted is given by Hooke's Law: \\[ F_{spring} = k imes x \] \where \( k \) is the spring constant, and \( x \) is the displacement. However, since this problem begins with the spring unstressed and focuses on the external force \( F \), we examine how the spring balances the forces within the system. When the blocks accelerate together, the spring exerts forces internally to maintain the motion dictated by the spring鈥檚 nature and the external force.

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Most popular questions from this chapter

A body of mass \(m\) is suspended by two strings making angle \(\alpha\) and \(\beta\) with the horizontal as shown in figure. Tensions in the two strings are (a) \(T_{1}=\frac{m g \cos \beta}{\sin (\alpha+\beta)}=T_{2}\) (b) \(T_{1}=\frac{m g \sin \beta}{\sin (\alpha+\beta)}=T_{2}\) (c) \(T_{1}=\frac{m g \cos \beta}{\sin (\alpha+\beta)} ; T_{1}=\frac{m g \cos \alpha}{\sin (\alpha+\beta)}\) (d) None of the above

A man weighing \(60 \mathrm{~kg}\) is standing on a trolley weighing \(240 \mathrm{~kg}\). The trolley is resting on frictionless horizontal rails. If the man starts walking on the trolley with a velocity of \(1 \mathrm{~ms}^{-1}\), then after \(4 \mathrm{~s}\), his displacement relative to the ground is (a) \(6 \mathrm{~m}\) (b) \(4.8 \mathrm{~m}\) (c) \(3.2 \mathrm{~m}\) (d) \(2.4 \mathrm{~m}\)

A force of \((5+3 x) \mathrm{N}\) acting on a body of mass \(20 \mathrm{~kg}\) along the \(x\)-axis displaces it from \(x=2 \mathrm{~m}\) to \(x=6 \mathrm{~m}\). The work done by the force is (a) \(20 \mathrm{~J}\) (b) \(48 \mathrm{~J}\) (c) 68\(]\) (d) \(86 \mathrm{~J}\)

If a force of \(250 \mathrm{~N}\) act on body the momentum acquired is \(125 \mathrm{~kg}-\mathrm{m} / \mathrm{s}\). What is the period for which force acts on the body? (a) \(0.5 \mathrm{~s}\) (b) \(0.2 \mathrm{~s}\) (c) \(0.4 \mathrm{~s}\) (d) \(0.25 \mathrm{~s}\)

Two weights \(w_{1}\) and \(w_{2}\) are suspended from the ends of a light string over a smooth fixed pulley. If the pulley is pulled up with acceleration \(g\), the tension in the string will be (a) \(\frac{4 w w_{2}}{w_{1}+w_{2}}\) (b) \(\frac{2 w w_{2}}{w_{1}+w_{2}}\) (c) \(\frac{w_{1}-w_{2}}{w_{1}+w_{j}}\) (d) \(\frac{w_{1} w_{2}}{2\left(w_{1}+w_{2}\right)}\)
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