/*! 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 50 A block of mass \(M\) slides on ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 50

A block of mass \(M\) slides on a thin film of oil. The film thickness is \(h\) and the area of the block is \(A\). When released, mass \(m\) exerts tension on the cord, causing the block to accelerate. Neglect friction in the pulley and air resistance. Develop an algebraic expression for the viscous force that acts on the block when it moves at speed \(V\). Derive a differential equation for the block speed as a function of time. Obtain an expression for the block speed as a function of time. The mass \(M=5 \mathrm{kg}, m=1 \mathrm{kg}, A=25 \mathrm{cm}^{2},\) and \(h=0.5 \mathrm{mm} .\) If it takes \(1 \mathrm{s}\) for the speed to reach \(1 \mathrm{m} / \mathrm{s}\), find the oil viscosity \(\mu .\) Plot the curve for \(V(t)\).

Short Answer

Expert verified
The equation for the force on the block is \((M+m) a = M g - \mu * A * V / h)\). The differential equation for speed is \((M+m) dv/dt = M g - \mu * A * V / h)\). The solution to this differential equation is \(V(t) = (Mg / \mu A / h) (1 - e^{-(\mu * A / h ) * t / M + m} )\). After solving for \(\mu\), we can plot the speed function \(V(t)\).

Step by step solution

01

Determine the Force on the Block

First, let's resolve the forces on the block. The gravity of block \(M\) pulls downwards with force \(Mg\). The tension in the string due to mass \(m\) pulls in the opposite direction with force \(mg\). As the block moves through the oil, it experiences a viscous force \(F\), given by the formula \(F = \mu * A * V / h\), where \(\mu\) is the viscosity of the oil, \(A\) is the contact area, \(V\) is the speed, and \(h\) is the oil film thickness.
02

Create a Differential Equation for Speed

Now, let's balance the forces according to Newton's second law \(F = ma\). The acceleration \(a\) can be substituted as \(dv/dt\), where \(v\) is the velocity and \(t\) is time. This gives the equation: \((M+m) a = M g - \mu * A * V / h)\), which simplifies to: \((M+m) dv/dt = M g - \mu * A * V / h)\), which is our differential equation for speed.
03

Solve the Differential Equation

Separate and integrate the differential equation. By doing so, the following expression for speed can be found: \(V(t) = (Mg / \mu A / h) (1 - e^{-(\mu * A / h ) * t / M + m} )\).
04

Find the Oil Viscosity

To find the oil viscosity, take the given condition: it takes 1 second to reach a speed of 1 m/s. Substituting these values and the given constants into the speed expression, one can solve for \(\mu\).
05

Plotting the Speed Function

The function \(V(t)\) can now be plotted over a reasonable range of time values using any graphing software. The function will show exponential growth, reaching an asymptote at \(V = \frac{Mg}{\mu A/h}\)

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Most popular questions from this chapter

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