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Date: 13-9-2020
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Date: 9-3-2021
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Date: 28-12-2016
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A pendulum is a very important type of oscillating motion and a very important clock (e.g. ''Grandfather Clock"). Let's analyze the forces and show that the period is independent of amplitude.
Example Prove that the period of a pendulum undergoing small oscillations is given by where L is the length of the pendulum.
Solution we have
where we take the x direction to be perpendicular to the string. Thus
where α is the angular acceleration . Now for small oscillations, sin θθ, so that
Now compare this to our spring equation which was
which had period . Thus for the pendulum we must have
Example Prove that the period of oscillation is , where I is the rotational inertia, m is the total mass and h is the distance from the rotation axis to the center of mass. Assume small oscillations.
Solution The torque is
where the minus sign indicates that when θ increases the torque acts in the opposite direction. For small oscillations sin θθ giving
Substitute into Newton's second law
gives
Now compare this to our spring equation which was
which had period . Thus for the physical pendulum we must have
FIGURE 1.1 Block sliding on frictionless surface with various spring combinations.
Example Two springs, with spring constants k1 and k2, are connected in parallel to a mass m sliding on a frictionless surface, as shown in Fig. 1.1a. What is the effective spring constant K? (i.e. If the two springs were replaced by a single spring with constant K, what is K in terms of k1 and k2?) Assume both springs have zero mass.
Solution If m moves by an amount x then it feels two forces -k1x and -k2x, giving
giving
Example The two springs of the previous example are connected in series, as shown in Fig. 1.1b. What is the effective spring constant K ?
Solution If spring 1 moves a distance x1 and spring 2 moves a distance x2 then the mass moves a distance x1 + x2. The force the mass feels is
Now consider the motion of the mass plus spring 2 system. The force it feels is
but we must have F = f because ma is same for mass m and mass plus spring 2 system because spring 2 has zero mass. Thus
but
(the ratio of stretching is inversely proportional to spring strength.) Thus giving
or
Example The two springs of the previous example are connected as shown in Fig.1.1c. What is the effective spring constant K ?
Solution If spring 1 is compressed by x then spring 2 is stretched by -x. Thus
giving
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تفوقت في الاختبار على الجميع.. فاكهة "خارقة" في عالم التغذية
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أمين عام أوبك: النفط الخام والغاز الطبيعي "هبة من الله"
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قسم شؤون المعارف ينظم دورة عن آليات عمل الفهارس الفنية للموسوعات والكتب لملاكاته
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