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Chapter 27: Problem 71

The bar in Problem 44 has mass \(m\) and is initially at rest. A constant force \(\vec{F}\) to the right is applicd to the bar. Formulate Newton's second law for the bar, and find its velocity as a function of time.

Short Answer

Expert verified
The velocity of the bar as a function of time is given by \(V = (\vec{F}/m) \cdot T\)

Step by step solution

01

Formulate Newton's Second Law

In accordance with Newton's second law, the net force acting on the bar is equal to the mass of the bar times its acceleration. In mathematical terms, this can be represented as: \(\vec{F} = m \cdot a\), where \(\vec{F}\) is the force, \(m\) is the mass and \(a\) is the acceleration of the bar.
02

Express Acceleration in Terms of Force and Mass

From the above equation, acceleration of the bar can be represented as \(a = \vec{F}/m\). Since force is a constant with respect to time, acceleration is therefore, also a constant with respect to time.
03

Find Velocity as a Function of Time

Since acceleration \(a\) is the derivative of velocity \(v\) with respect to time \(t\), we get \(a = dv/dt\). Substituting for \(a\) gives \(dv/dt = \vec{F}/m\). Rearranging terms and integrating from \(v=0\) at \(t=0\) and \(v=V\) at \(t=T\), you find the integral \(\int_{0}^{T} dv = \int_{0}^{T} \vec{F}/m dt\). Evaluating these integrals gives \(V = (\vec{F}/m) \cdot T\), which is the velocity of the bar as a function of time.

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