Explanation Take the point of contact of the sign with the wall as the rotation point. The two torques acting on the sign about this rotation point are the torque due to the weight of the sign and the torque due to the tension in the rope. The sign is in equilibrium so that the net torque must be zero. This implies the two torques acting on the sign about the rotation point must be equal and opposite. Write the equation for the net torque.00000 τ → = τ → g + τ → T Here, τ → is the net torque, τ → g is the torque due to the weight of the sign and τ → T is the torque due to the tension in the rope. Equate the above equation to zero. τ → g + τ → T = 0 | τ → g | = | τ → T | (I) The diagram is shown in figure 1. Write the equation for torque. τ → = r → × F → (II) Here, τ is the torque, r → is the vector representing the perpendicular distance from the axis of rotation to the point of action of the force and F → is the force. Use equation (II) to write the expression for the torque due to the weight. τ → g = r → g × F → g (III) Here,, r → g is the perpendicular distance between the axis of rotation and the point where the weight acts and F → g is the weight. Write the equation for F → g . F → g = − m g j ^ Here, m is the mass of the sign and g is the acceleration due to gravity. The perpendicular distance between the weight and the axis of rotation is the radius of the circular sign. Refer to figure 1 and write the equation for r → g . r → g = R i ^ Here, R is the radius of the circular sign. Put the above two equations in equation (III). τ → g = ( R i ^ ) × ( − m g j ^ ) = − m g R k ^ (IV) Use equation (I) to write the expression for the torque due to the tension in the rope. τ → T = r → × F → T (V) Here, r → is the vector denoting the perpendicular distance between the axis of rotation and the point where the tension acts and F → T is the tension. Refer to figure 1 and write the equation for F → T . F → T = − F T sin 60 ° i ^ + F T cos 60 ° j ^ = − 3 2 F T i ^ + 1 2 F T j ^ (VI) Here, F T is the magnitude of the tension

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN: 9781337026345

Author: Katz

Publisher: Cengage

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Physics for Scientists and Engineers: Foundations and Connections

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The tension in the cable.

Solution Summary: The author analyzes the tension in the cable. Take the point of contact of the sign with the wall as the rotation point.

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Chapter 14 Solutions