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Chapter 7: Problem 38

Prove Eq. $(7-13) \Sigma \overrightarrow{\mathbf{F}}_{\mathrm{ext}}=M \overrightarrow{\mathbf{a}}_{\mathrm{CM}} .\( [Hint: Start with \)\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{cxt}}=\lim _{\Delta t \rightarrow 0}(\Delta \overrightarrow{\mathbf{p}} / \Delta t),\( where \)\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{ext}}$ is the net external force acting on a system and \(\overrightarrow{\mathbf{p}}\) is the total momentum of the system.]

Short Answer

Expert verified
Question: Prove that the net external force acting on a system is equal to the product of the total mass of the system and the acceleration of its center of mass, given the expression \(\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{cxt}}=\lim _{\Delta t \rightarrow 0}(\Delta \overrightarrow{\mathbf{p}} / \Delta t)\). Answer: By defining the total momentum of the system, expressing the change in momentum, defining acceleration, substituting the change in velocity, using the limit, and simplifying the expression, we obtain \(\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{cxt}} = M \overrightarrow{\mathbf{a}}_{\mathrm{CM}}\), which proves the required statement.

Step by step solution

01

Start with the given formula

We are provided with the formula \(\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{cxt}}=\lim _{\Delta t \rightarrow 0}(\Delta \overrightarrow{\mathbf{p}} / \Delta t)\), which expresses the net external force acting on a system in terms of the change in the total momentum of the system over an infinitesimally small time interval.
02

Express the change in momentum

Recall that the total momentum of a system is given by \(\overrightarrow{\mathbf{p}} = M\overrightarrow{\mathbf{v}}_{\mathrm{CM}}\), where \(M\) is the total mass of the system and \(\overrightarrow{\mathbf{v}}_{\mathrm{CM}}\) is the velocity of the center of mass. To find the change in momentum, we can write \(\Delta \overrightarrow{\mathbf{p}} = M\Delta \overrightarrow{\mathbf{v}}_{\mathrm{CM}}\).
03

Define acceleration

Acceleration is defined as the change in velocity over time. Therefore, we can express the change in the center of mass velocity as \(\Delta \overrightarrow{\mathbf{v}}_{\mathrm{CM}} = \overrightarrow{\mathbf{a}}_{\mathrm{CM}}\Delta t\).
04

Substitute the change in velocity

Now, substitute the expression for the change in velocity from step 3 into the change in momentum from step 2: \(\Delta \overrightarrow{\mathbf{p}} = M(\overrightarrow{\mathbf{a}}_{\mathrm{CM}}\Delta t)\).
05

Use the limit

Now use the limit to find the net external force: \(\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{cxt}} = \lim _{\Delta t \rightarrow 0}(\Delta \overrightarrow{\mathbf{p}} / \Delta t) = \lim _{\Delta t \rightarrow 0}\frac{M(\overrightarrow{\mathbf{a}}_{\mathrm{CM}}\Delta t)}{\Delta t}\).
06

Simplify the expression

Cancel the \(\Delta t\) terms in the numerator and the denominator: \(\Sigma \overrightarrow{\mathbf{F}}_{\mathrm{cxt}} = M \overrightarrow{\mathbf{a}}_{\mathrm{CM}}\). This expression shows that the net external force acting on a system is equal to the product of the total mass of the system and the acceleration of its center of mass, as required.

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