Concept explainers
(a)
The length of the spring.
(a)
Answer to Problem 45PQ
The length of the spring is
Explanation of Solution
Following figure gives rod and spring system.
Following figure is the free body diagram of rod and spring system.
Write the expression for the horizontal distance between one end to other end of the spring.
Here,
Write the expression for
Here,
Write the expression for the vertical distance of the end of the spring from ground.
Here,
Write the expression for the length of the spring using Pythagoras theorem.
Here,
Conclusion:
Substitute
Substitute
This distance is also equal to vertical distance of the end of the spring from ground.
Substitute
Substitute
Therefore, the length of the spring is
(b)
The weight of the bar.
(b)
Answer to Problem 45PQ
The weight of the bar is
Explanation of Solution
At equilibrium, the net torque acting on the bar around the bottom pivot must be zero.
Write the expression for the torque about pivot in the rod due to gravity.
Here,
The direction of torque is into the page.
Using figure2, write the expression for the perpendicular distance between pivot of the rod and point where weight acts.
Write the expression for the radial vector from the pivot point to the end of the bar where the spring acts.
Here,
Write the expression for the relaxed length.
Here,
The magnitude of spring force depends on the extension relative to the relaxed spring length.
Write the expression for the magnitude of spring force.
Here,
Write the expression for the extension of spring relative to the relaxed spring length.
From figure2, write the expression for the angle spring force makes below the horizontal.
Here,
Write the expression for the spring force as a vector.
Here,
Write the expression for the torque on the rod due to spring force.
Here,
The direction of above torque is out of the page.
At equilibrium torque due to spring force and weight will cancel each other. Since, the directions of torques are opposite, their magnitude should be equal.
Write the equilibrium condition of the torques.
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Use equations (XV) and (XVI) in (XIII) to get
Substitute
Therefore, The weight of the bar is
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Chapter 14 Solutions
Physics for Scientists and Engineers: Foundations and Connections
- Three forces are exerted on the disk shown in Figure P12.71,and their magnitudes are F3 = 2F2 = 2F1. The disks outer rimhas radius R, and the inner rim has radius R/2. As shown in thefigure, F1 and F3 are tangent to the outer rim of the disk, and F2 is tangent to the inner rim. F3 is parallel to the x axis, F2 is parallel to the y axis, and F1 makes a 45 angle with the negative x axis. Find expressions for the magnitude of each torque exertedaround the center of the disk in terms of R and F1. FIGURE P12.71 Problems 71-75arrow_forwardA smaller disk of radius r and mass m is attached rigidly to the face of a second larger disk of radius R and mass M as shown in Figure P12.64. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is v=2[Rg(1cos)(M/m)+(r/R)2+2]1/2 (b) Show that the period of the motion is T=2[(M+2m)R2+mr22mgR]1/2 Figure P12.64arrow_forwardReview. One end of a light spring with force constant k = 100 N/m is attached to a vertical wall. A light string is tied to the other end of the horizontal spring. As shown in Figure P12.57, the string changes from horizontal to vertical as it passes over a pulley of mass M in the shape of a solid disk of radius R = 2.00 cm. The pulley is free to turn on a fixed, smooth axle. The vertical section of the string supports an object of mass m = 200 g. The string does not slip at its contact with the pulley. The object is pulled downward a small distance and released. (a) What is the angular frequency of oscillation of the object in terms of the mass M? (b) What is the highest possible value of the angular frequency of oscillation of the object? (c) What is the highest possible value of the angular frequency of oscillation of the object if the pulley radius is doubled to R = 4.00 cm? Figure P12.57arrow_forward
- A horizontal, rigid bar of negligible weight is fixed against a vertical wall at one end and supported by a vertical string at the other end. The bar has a length of 50.0 cm and is used to support a hanging block of weight 400.0 N from a point 30.0 cm from the wall as shown in Figure P14.81. The string is made from a material with a tensile strength of 1.2 108 N/m2. Determine the largest diameter of the string for which it would still break. FIGURE P14.81arrow_forwardA uniform rod of weight Fg and length L is supported at its ends by a frictionless trough as shown in Figure P12.49. (a) Show that the center of gravity of the rod must be vertically over point O when the rod is in equilibrium. (b) Determine the equilibrium value of the angle . (c) Is the equilibrium of the rod stable or unstable? Figure P12.49arrow_forwardA spring with spring constant 25 N/m is compressed a distance of 7.0 cm by a ball with a mass of 202.5 g (Fig. P13.33). The ball is then released and rolls without slipping along a horizontal surface, leaving the spring at point A. The process is repeated, using a block instead, with a mass identical to that of the ball. The block compresses the spring by 7.0 cm and is also released, leaving the spring at point A. Assume the ball rolls, but ignore other effects of friction. a. What is the speed of the ball at point B? b. What is the speed of the block at point B? FIGURE P13.33 Problems 33 and 34.arrow_forward
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