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Chapter 10: Problem 44

A force, \(\vec{F}=(2 \hat{x}+3 \hat{y}) \mathrm{N},\) is applied to an object at a point whose position vector with respect to the pivot point is \(\vec{r}=(4 \hat{x}+4 \hat{y}+4 \hat{z}) \mathrm{m} .\) Calculate the torque created by the force about that pivot point.

Short Answer

Expert verified
Question: Calculate the torque created by the force \(\vec{F} = (2\hat{x} + 3\hat{y})\,\text{N}\) applied at a point with the position vector \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})\,\text{m}\) with respect to the pivot point. Answer: The torque is \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\).

Step by step solution

01

Write down the given vectors

We are given the force vector \(\vec{F} = (2\hat{x} + 3\hat{y})\,\text{N}\) and the position vector \(\vec{r} = (4\hat{x} + 4\hat{y} + 4\hat{z})\,\text{m}\).
02

Calculate the cross product of \(\vec{r}\) and \(\vec{F}\)

To find the torque vector \(\vec{\tau}\), we need to calculate the cross product of \(\vec{r}\) and \(\vec{F}\). The cross product of two vectors is given by: \(\vec{\tau} = \vec{r} \times \vec{F} = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ 4 & 4 & 4 \\ 2 & 3 & 0 \end{vmatrix}\)
03

Evaluate the cross product

Now, let's evaluate the cross product using the formula: \(\vec{\tau} = (\hat{x}(4\cdot0 - 3\cdot4) - \hat{y}(4\cdot0 - 2\cdot4) + \hat{z}(4\cdot3 - 2\cdot4))\) \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\)
04

Write the final answer

The torque created by the force \(\vec{F}\), about the pivot point with the position vector \(\vec{r}\), is: \(\vec{\tau} = (-12\hat{x} + 8\hat{y} + 4\hat{z})\,\text{Nm}\)

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Most popular questions from this chapter

A wheel with \(c=\frac{4}{9}\), a mass of \(40.0 \mathrm{~kg}\), and a rim radius of \(30.0 \mathrm{~cm}\) is mounted vertically on a horizontal axis. A 2.00 -kg mass is suspended from the wheel by a rope wound around the rim. Find the angular acceleration of the wheel when the mass is released.

You are the technical consultant for an action-adventure film in which a stunt calls for the hero to drop off a 20.0 -m-tall building and land on the ground safely at a final vertical speed of \(4.00 \mathrm{~m} / \mathrm{s}\). At the edge of the building's roof, there is a \(100 .-\mathrm{kg}\) drum that is wound with a sufficiently long rope (of negligible mass), has a radius of \(0.500 \mathrm{~m}\), and is free to rotate about its cylindrical axis with a moment of inertia \(I_{0}\). The script calls for the 50.0 -kg stuntman to tie the rope around his waist and walk off the roof. a) Determine an expression for the stuntman's linear acceleration in terms of his mass \(m\), the drum's radius \(r\) and moment of inertia \(I_{0}\). b) Determine the required value of the stuntman's acceleration if he is to land safely at a speed of \(4.00 \mathrm{~m} / \mathrm{s},\) and use this value to calculate the moment of inertia of the drum about its axis. c) What is the angular acceleration of the drum? d) How many revolutions does the drum make during the fall?

A uniform rod of mass \(M=250.0 \mathrm{~g}\) and length \(L=50.0 \mathrm{~cm}\) stands vertically on a horizontal table. It is released from rest to fall. a) What forces are acting on the rod? b) Calculate the angular speed of the rod, the vertical acceleration of the moving end of the rod, and the normal force exerted by the table on the rod as it makes an angle \(\theta=45.0^{\circ}\) with respect to the vertical. c) If the rod falls onto the table without slipping, find the linear acceleration of the end point of the rod when it hits the table and compare it with \(g\).

A thin uniform rod (length \(=1.00 \mathrm{~m},\) mass \(=2.00 \mathrm{~kg})\) is pivoted about a horizontal frictionless pin through one of its ends. The moment of inertia of the rod through this axis is \(\frac{1}{3} m L^{2} .\) The rod is released when it is \(60.0^{\circ}\) below the horizontal. What is the angular acceleration of the rod at the instant it is released?

In another race, a solid sphere and a thin ring roll without slipping from rest down a ramp that makes angle \(\theta\) with the horizontal. Find the ratio of their accelerations, \(a_{\text {ring }} / a_{\text {sphere }}\)
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