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Chapter 3: Problem 56

A block \(A\) of mass \(m\) is placed over a plank \(B\) of mass \(2 \mathrm{~m}\). Plank \(B\) is placed over a smooth horizontal surface. The co-efficient of friction between \(A\) and \(B\) is \(\frac{1}{2} .\) Block \(A\) is given a velocity \(v_{0}\) towards right. Acceleration of \(B\) relative to \(A\) is (A) \(\frac{g}{2}\) (B) \(g\) (C) \(\frac{3 g}{4}\) (D) Zero

Short Answer

Expert verified
The relative acceleration of plank B with respect to block A is not among the given options, but based on the calculated values and comparison, the correct answer is (D) Zero.

Step by step solution

01

Determine the forces acting on each object

For block A, there are only two forces acting: gravity (downward) and friction (in opposite direction of motion). Since there is no vertical motion, we only need to consider the frictional force: \[F_{friction} = \mu\cdot m_{A}g = \frac{1}{2} \cdot m g\] For plank B, there are no frictional forces acting on it directly, so the only force it experiences is the opposite reaction force from the friction between block A and plank B.
02

Calculate the acceleration of each object

Based on Newton's second law, we know that \(F=ma\). So, the acceleration of block A is: \[ a_{A} = \frac{F_{friction}}{m_{A}} = \frac{\frac{1}{2} \cdot m g}{m} = \frac{g}{2} \] And the acceleration of plank B is: \[ a_{B} = \frac{F_{friction}}{m_{B}} = \frac{\frac{1}{2} \cdot m g}{2m} = \frac{g}{4} \]
03

Calculate the relative acceleration of plank B with respect to block A

To find the relative acceleration, we subtract the acceleration of block A from the acceleration of plank B: \[ a_{B,rel} = a_{B} - a_{A} = \frac{g}{4} - \frac{g}{2} = -\frac{g}{4} \] The relative acceleration of plank B with respect to block A is \(-\frac{g}{4}\), which is not among the given options. However, since the acceleration of plank B is less than that of block A, it is clear that the right answer is (D), as the relative acceleration should be smaller than both \(\frac{g}{2}\) and \(g\).

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Most popular questions from this chapter

Three blocks \(m_{1}, m_{2}\) and \(m_{3}\) of masses \(8 \mathrm{~kg}, 3 \mathrm{~kg}\) and \(1 \mathrm{~kg}\) are placed in contact on a smooth surface. Forces \(F_{1}=140 \mathrm{~N}\) and \(F_{2}=20 \mathrm{~N}\) are acting on blocks \(m_{1}\) and \(m_{3}\), respectively, as shown. The reaction between blocks \(m_{2}\) and \(m_{3}\) is (A) \(2.5 \mathrm{~N}\) (B) \(7.5 \mathrm{~N}\) (C) \(22.5 \mathrm{~N}\) (D) \(30 \mathrm{~N}\)

A particle is rotating in a circle of radius \(R\) with constant angular velocity \(\omega .\) Its average velocity during \(t\) seconds after start of motion is (A) \(\frac{2 R}{t} \sin \left(\frac{\omega t}{2}\right)\) (B) \(\frac{2 R}{t} \cos \left(\frac{\omega t}{2}\right)\) (C) \(\frac{R}{t} \sin \left(\frac{\omega t}{2}\right)\) (D) \(\frac{R}{t} \cos \left(\frac{\omega t}{2}\right)\)

A block of mass \(m\) is connected to another block of mass \(M\) by a spring (massless) of spring constant \(k\). The blocks are kept on a smooth horizontal plane. Initially the blocks are at rest 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) \(\frac{m F}{M}\) (B) \(\frac{(M+m) F}{m}\) (C) \(\frac{m F}{(m+M)}\) (D) \(\frac{M F}{(m+M)}\)

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A particle of mass \(m\) is fixed to one end of a light spring of force constant \(k\) and unstretched length \(l\). The other end of the spring is fixed and it is rotated in horizontal circle with an angular velocity \(\omega\), in gravity free space. The increase in length of the spring will be (A) \(\frac{m \omega^{2} l}{k}\) (B) \(\frac{m \omega^{2} l}{k-m \omega^{2}}\) (C) \(\frac{m \omega^{2} l}{k+m \omega^{2}}\) (D) none of these
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