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Chapter 11: Q39P (page 463)

(a) What is the period of small-angle oscillations of a simple pendulum with a mass of 0.1kgat the end of a string of length1m?(b) What is the period of small-angle oscillations of a meter stick suspended from one end, whose mass is0.1kg?

Short Answer

Expert verified

The period of small angle oscillations of a sample pendulum is 2.01 s.

The period of small angle oscillation of a meter stick suspended from one end is 1.64 s.

Step by step solution

01

Define oscillations of a simple pendulum.

A pendulum completes one oscillation when it begins at one extreme position A, goes to the opposite extreme point B, and then returns to A. The time period is the amount of time it takes to complete one oscillation. The oscillation's time period remains constant.

02

Find the period of small angle oscillations of a simple pendulum.

  1. The period of small angle oscillations of a sample pendulum is,

T=2蟺lg

Here, lis length of the pendulum and gis acceleration due to gravity.

Substitute 1mfor land 9.8m/s2for g.

T=2蟺1m9.8m/s2=2.01s

Therefore, the period of small angle oscillations of a simple pendulum is2.01s.

03

Find the period of small angle oscillation of a meter stick suspended from one end.

b. The period of small angle oscillation of a meter stick suspended from one end is,


T=2蟺Imgl'


Here,l'is length of stick from its center to the one end,mis mass of the scale and lis moment of inertia of the scale relative to one end.

The moment of inertia of the scale is,

I=13ml2


Substitute 13ml2for Iand l2for l'in equation T=2蟺Imgl'


T=2蟺13ml2mgl2=2蟺2l3g


Substitute 1mfor land 9.8m/s2for g.


T=2蟺21m39.8m/s2=1.64s

Therefore, the period of small angle oscillation of a meter stick suspended from one end is1.64s.

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Most popular questions from this chapter

A stationary bicycle wheel of radius 0.9mis mounted in the vertical plane (figure). The axle is held up by supports that are not shown, and the wheel is free to rotate on the nearly frictionless axle. The wheel has mass all 4.8kgconcentrated in the rim (the spokes have negligible mass). A lump of clay with mass 0.5kgfalls and sticks to the outer edge of the wheel at the location shown. Just before the impact the clay has speed 5m/sand the wheel is rotating clockwise with angular speed0.33rad/s.

(a) Just before the impact, what is the angular momentum (magnitude and direction) of the combined system of wheel plus clay about the center C? (As usual, is to the right, is up, and is out of the screen, towards you) (b) Just after the impact, what is the angular momentum (magnitude and direction) of the combined system of wheel plus clay about the center C? (c) Just after the impact, what is the angular velocity (magnitude and direction) of the wheel? (d) Qualitatively, what happens to the linear momentum of the combined system? Why? (1) The downward linear momentum decreases because the axle exerts into angular momentum. (3) some of the linear momentum is changed into energy. (4) There is no change because linear momentum is always conserved.

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In Figure 11.95 two small objects each of mass \({m_1}\)are connected by a light weight rod of length \(L.\) At a particular instant the center of mass speed is\({v_1}\) as shown, and the object is rotating counterclockwise with angular speed \({\omega _1}\). A small object of mass \({m_2}\) travelling with speed \({v_2}\) collides with the rod at an angle \({\theta _2}\) as shown, at a distance\(b\)from the center of the rod. After being truck, the mass \({m_2}\) is observed to move with speed \({v_4}\) at angle\({\theta _4}\).All the quantities are positive magnitudes. This all takes place in outer space.

For the object consisting of the rod with the two masses, write equations that, in principle, could be solved for the center of mass speed \({v_3},\) direction \({\theta _3},\) and angular speed \({\omega _3}\)in terms of the given quantities. Sates clearly what physical principles you use to obtain your equations.

Don鈥檛 attempt to solve the equations; just set them up.

Under what conditions is the torque about some location equal to zero?

Determine both the direction and magnitude of the angular momentum of the particle in Figure 11.13, relative to the locations D, E,F, G, and H. We've already analyzed the angular momentum relative toA, B, and C in the example given above. Notice how the magnitude and direction of the angular momentum relative to the different locations differ in magnitude and direction.
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