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Chapter 10: Problem 68

A 0.050 -kg bead slides on a wire bent into a circle of radius \(0.40 \mathrm{~m}\). You pluck the bead with a force tangent to the circle. What force is needed to give the bead an angular acceleration of \(6.0 \mathrm{rad} / \mathrm{s}^{2} ?\)

Short Answer

Expert verified
Answer: The force needed to achieve the given angular acceleration is 0.12 N.

Step by step solution

01

Write down Newton's second law for rotational motion

Newton's second law for rotational motion is given by: \(\tau = I \alpha\) Where \(\tau\) is the torque, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration.
02

Find the moment of inertia of the bead

Since the bead is moving on a circular path with a constant radius, we can treat it as a point mass and use the formula for the moment of inertia of a point mass: \(I = mr^2\) where \(m\) is the mass of the bead, and \(r\) is the radius of the circle. Plugging in the given values, we get: \(I = (0.050\,\text{kg})(0.40\,\text{m})^2 = 0.008\,\text{kg}\cdot\text{m}^2\)
03

Calculate the torque needed to achieve the given angular acceleration

Now that we have the moment of inertia, we can use Newton's second law for rotational motion to find the torque needed to achieve the given angular acceleration: \(\tau = I \alpha\) Plugging in the values for \(I\) and \(\alpha\), we get: \(\tau = (0.008\,\text{kg}\cdot\text{m}^2)(6.0\,\text{rad}/\text{s}^2) = 0.048\,\text{N}\cdot\text{m}\)
04

Determine the tangential force needed to produce the required torque

Since the force is applied tangentially to the bead, the torque produced by the force can be calculated as: \(\tau = r F_\text{t}\) where \(F_\text{t}\) is the tangential force. We can now solve for \(F_\text{t}\): \(F_\text{t} = \frac{\tau}{r}\) Substituting the values, we get: \(F_\text{t} = \frac{0.048\,\text{N}\cdot\text{m}}{0.40\,\text{m}} = 0.12\,\text{N}\) Therefore, the force needed to give the bead an angular acceleration of \(6.0\,\text{rad}/\text{s}^2\) is \(0.12\,\text{N}\).

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Most popular questions from this chapter

A solid sphere of radius \(R\) and mass \(M\) is placed at a height \(h_{0}\) on an inclined plane of slope \(\theta\). When released, it rolls without slipping to the bottom of the incline. Next, a cylinder of same mass and radius is released on the same incline. From what height \(h\) should it be released in order to have the same speed as the sphere at the bottom?

A round body of mass \(M\), radius \(R,\) and moment of inertia \(I\) about its center of mass is struck a sharp horizontal blow along a line at height \(h\) above its center (with \(0 \leq h \leq R,\) of course). The body rolls away without slipping immediately after being struck. Calculate the ratio \(I /\left(M R^{2}\right)\) for this body.

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