# Daily Archives: August 16, 2019

573 posts

## 14.96 By what percentage does the frequency of oscillation change if ksurf = 5 N>m? (a) 0.1%; (b) 0.2%; (c) 0.5%; (d) 1.0%.

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## 14.95 In the model of Problem 14.94 what is the mechanical energy of the vibration when the tip is not interacting with the surface? (a) 1.2 * 10-18 J; (b) 1.2 * 10-16 J; (c) 1.2 * 10-9 J; (d) 5.0 * 10-8 J.

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## 14.94 If we model the vibrating system as a mass on a spring what is the mass necessary to achieve the desired resonant frequency when the tip is not interacting with the surface? (a) 25 ng; (b) 100 ng; (c) 2.5 mg; (d) 100 mg.

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## 14.93 … CALC A Spring with Mass. The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the spring’s mass consider a spring with mass M equilibrium length L0 and spring constant k. When stretched or compressed to a length L the potential energy is 12 kx2 where x = L – L0. (a) Consider a spring as described above that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v. (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l v and L; find the mass of each piece in terms of dl M and L; and integrate from 0 to L. The result is not 12 Mv2 since not all of the spring moves with the same speed.) (b) Take the time derivative of the conservation of energy equation Eq. (14.21) for a mass m moving on the end of a massless spring. By comparing your results to Eq. (14.8) which defines v show that the angular frequency of oscillation is v = 1k>m. (c) Apply the procedure of part (b) to obtain the angular frequency of oscillation v of the spring considered in part (a). If the effective mass M_ of the spring is defined by v = 1k>M_ what is M_ in terms of M?

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## 14.92 … The Effective Force Constant of Two Springs. Two springs with the same unstretched length but different force constants k1 and k2 are attached to a block with mass m on a level frictionless surface. Calculate the effective force constant keff in each of the three cases (a) (b) and (c) depicted in Fig. P14.92. (The effective force constant is defined by gFx = -keff x.) (d) An object with mass m suspended from a uniform spring with a force constant k vibrates with a frequency f1. When the spring is cut in half and the same object is suspended from one of the halves the frequency is f2. What is the ratio f1>f2?

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## 14.91 … DATA Experimenting with pendulums you attach a light string to the ceiling and attach a small metal sphere to the lower end of the string. When you displace the sphere 2.00 m to the left it nearly touches a vertical wall; with the string taut you release the sphere from rest. The sphere swings back and forth as a simple pendulum and you measure its period T. You repeat this act for strings of various lengths L each time starting the motion with the sphere displaced 2.00 m to the left of the vertical position of the string. In each case the sphere’s radius is very small compared with L. Your results are given in the table: L 1m2 12.00 10.00 8.00 6.00 5.00 4.00 3.00 2.50 2.30 T 1s 2 6.96 6.36 5.70 4.95 4.54 4.08 3.60 3.35 3.27 (a) For the five largest values of L graph T2 versus L. Explain why the data points fall close to a straight line. Does the slope of this line have the value you expected? (b) Add the remaining data to your graph. Explain why the data start to deviate from the straight-line fit as L decreases. To see this effect more clearly plot T>T0 versus L where T0 = 2p1L>g and g = 9.80 m>s2. (c) Use your graph of T>T0 versus L to estimate the angular amplitude of the pendulum (in degrees) for which the equation T = 2p1L>g is

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## 14.90 .. DATA You hang various masses m from the end of a vertical 0.250-kg spring that obeys Hooke’s law and is tapered which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible you must replace m in the equation T = 2p1m>k with m + meff where meff is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass m and measure the time for 10 complete oscillations obtaining these data: (a) Graph the square of the period T versus the mass suspended from the spring and find the straight line of best fit. (b) From the slope of that line determine the force constant of the spring. (c) From the vertical intercept of the line determine the spring’s effective mass. (d) What fraction is meff of the spring’s mass? (e) If a 0.450-kg mass oscillates on the end of the spring find its period frequency and angular frequency.

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## 14.89 .. DATA A mass m is attached to a spring of force constant 75 N/m and allowed to oscillate. Figure P14.89 shows a graph of its velocity component vx as a function of time t. Find (a) the period (b) the frequency and (c) the angular frequency of this motion. (d) What is the amplitude (in cm) and at what times does the mass reach this position? (e) Find the maximum acceleration magnitude of the mass and the times at which it occurs. (f) What is the value of m?

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## 14.88 … Two identical thin rods each with mass m and length L are joined at right angles to form an L-shaped object. This object is balanced on top of a sharp edge (Fig. P14.88). If the L-shaped object is deflected slightly it oscillates. Find the frequency of oscillation.

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## 14.87 .. CALC A slender uniform metal rod with mass M is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant k is attached to the lower end of the rod with the other end of the spring attached to a rigid support. If the rod is displaced by a small angle _ from the vertical (Fig. P14.87) and released show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle _ is small enough for the approximations sin _ _ _ and cos _ _ 1 to be valid. The motion is simple harmonic if d2u>dt2 = -v2u and the period is then T = 2p>v.)

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## 14.86 .. The Silently Ringing Bell. A large 34.0-kg bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell’s center of mass is 0.60 m below the pivot. The bell’s moment of inertia about an axis at the pivot is 18.0 kg # m2. The clapper is a small 1.8-kg mass attached to one end of a slender rod of length L and negligible mass. The other end of the rod is attached to the inside of the bell; the rod can swing freely about the same axis as the bell. What should be the length L of the clapper rod for the bell to ring silently—that is for the period of oscillation for the bell to equal that of the clapper?

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## 14.85 . CP In Fig. P14.85 the upper ball is released from rest collides with the stationary lower ball and sticks to it. The strings are both 50.0 cm long. The upper ball has mass 2.00 kg and it is initially 10.0 cm higher than the lower ball which has mass 3.00 kg. Find the frequency and maximum angular displacement of the motion after the collision.

University Physics with Modern Physics lessons by JJtheTutor. Designed to teach students problem solving skills, test taking skills and how to understand the concepts. ## Physics – Mechanics: Motion In One-Dimensions (21 of 22) Two Objects

Visit http://ilectureonline.com for more math and science lectures! In this video I will show you how to calculate the height when 2 objects will pass each other – one object thrown upward and the other dropped (free-fall) one second later. ## Horizontally launched projectile | Two-dimensional motion | Physics | Khan Academy

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Motion of a particle in three dimensions,acccelaration of a particle in terms of polar coordinate, Bsc mechanics motion in three dimension , Bsc clasess machines, Bsc dynamics and statics, Bsc mathematics, ## Physics – Mechanics: Motion In Two-Dimensions: (17 of 21) Circular Motion and Acceleration

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