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Chapter 9: Problem 1

A particle is located at coordinates \(x=2.0 \mathrm{~m}, y=3.0 \mathrm{~m}\) What is the magnitude of the torque about the origin when the particle is acted upon by a force of magnitude \(5.0 \mathrm{~N}\) in \((a)\) the positive \(x\) direction, \((b)\) the positive \(y\) direction, and \((c)\) the negative \(x\) direction?

Short Answer

Expert verified
The magnitudes of the torque are 0 N.m along both the positive and negative x directions, and 5 sqrt(13) N.m along the positive y direction.

Step by step solution

01

Calculate the Position Vector's Magnitude

The magnitude of the position vector r = sqrt((2.0m)^2 + (3.0m)^2) = sqrt(4 + 9) = sqrt(13) m.
02

Compute the Torque for Different Force Directions

Now, for each force directions, the magnitude of the torque τ = |rF sinΘ| will be computed. Here, r = sqrt(13)m and F = 5.0N. For (a) the positive x direction, Θ is the angle with x axis which is 0. Hence, τ = rFsin0 = 0 N.m. For (b) the positive y direction, Θ is the angle with y axis which is 90°. Hence, τ = rFsin90 = sqrt(13)m . 5.0N = 5 sqrt(13) N.m. For (c) the negative x direction, Θ is the angle with negative x axis which is 180°. Hence, τ=rFsin180 = 0 N.m.
03

Conclusion

Therefore, the magnitude of the torque about the origin when the particle is acted upon by a force is 0 N.m along the positive x and negative x directions, and 5 sqrt(13) N.m along the positive y direction.

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Most popular questions from this chapter

A particle is located at \(\overrightarrow{\mathbf{r}}=(0.54 \mathrm{~m}) \hat{\mathbf{i}}+(-0.36 \mathrm{~m}) \hat{\mathbf{j}}+\) \((0.85 \mathrm{~m}) \hat{\mathbf{k}}\). A constant force of magnitude \(2.6 \mathrm{~N}\) acts on the particle. Find the components of the torque about the origin when the force acts in \((a)\) the positive \(x\) direction and \((b)\) the negative \(z\) direction.

Two identical blocks, each of mass \(M\), are connected by a light string over a frictionless pulley of radius \(R\) and rotational inertia \(I\) (Fig. 9-55). The string does not slip on the pulley, and it is not known whether or not there is friction between the plane and the sliding block. When this system is released, it is found that the pulley turns through an angle \(\theta\) in time \(t\) and the acceleration of the blocks is constant. (a) What is the angular acceleration of the pulley? ( \(b\) ) What is the acceleration of the two blocks? ( \(c\) ) What are the tensions in the upper and lower sections of the string? All answers are to be expressed in terms of \(M, I, R, \theta, g\), and \(t\).

A 160 -lb person is walking across a level bridge and stops three-fourths of the way from one end. The bridge is uniform and weighs 600 lb. What are the values of the vertical forces exerted on each end of the bridge by its supports?

Each of three helicopter rotor blades shown in Fig. 9-42 is \(5.20 \mathrm{~m}\) long and has a mass of \(240 \mathrm{~kg}\). The rotor is rotating at 350 rev/min. What is the rotational inertia of the rotor assembly about the axis of rotation? (Each blade can be considered a thin rod.)

Calculate the rotational inertia of a meter stick, with mass \(0.56 \mathrm{~kg}\), about an axis perpendicular to the stick and located at the 20 -cm mark.
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