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A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension (Fig. P15.55). Find the frequency of vibration of the system for small values of the amplitude (small θ). Assume the vertical suspension rod of length L is rigid, but ignore its mass.
Figure P15.35
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Bundle: Physics for Scientists and Engineers with Modern Physics, Loose-leaf Version, 10th + WebAssign Printed Access Card for Serway/Jewett's Physics for Scientists and Engineers, 10th, Multi-Term
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- A block of mass m rests on a frictionless, horizontal surface and is attached to two springs with spring constants k1 and k2 (Fig. P16.22). It is displaced to the right and released. Find an expression for the angular frequency of oscillation of the resulting simple harmonic motion. FIGURE P16.22 Problems 22 and 81.arrow_forwardA small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P12.59). Determine the tensions in the rod (a) at the pivot and (b) at the point P when the system is stationary. (c) Calculate the period of oscillation for small displacements from equilibrium and (d) determine this period for L = 2.00 m. Figure P12.59arrow_forwardA watch balance wheel (Fig. P15.25) has a period of oscillation of 0.250 s. The wheel is constructed so that its mass of 20.0 g is concentrated around a rim of radius 0.500 cm. What are (a) the wheels moment of inertia and (b) the torsion constant of the attached spring? Figure P15.23arrow_forward
- A block of mass m is connected to two springs of force constants k1 and k2 in two ways as shown in Figure P12.56. In both cases, the block moves on a frictionless table after it is displaced from equilibrium and released. Show that in the two cases the block exhibits simple harmonic motion with periods (a) T=2m(k1+k2)k1k2 and (b) T=2mk1+k2 Figure P12.56arrow_forwardA spherical bob of mass m and radius R is suspended from a fixed point by a rigid rod of negligible mass whose length from the point of support to the center of the bob is L (Fig. P16.75). Find the period of small oscillation. N The frequency of a physical pendulum comprising a nonuniform rod of mass 1.25 kg pivoted at one end is observed to be 0.667 Hz. The center of mass of the rod is 40.0 cm below the pivot point. What is the rotational inertia of the pendulum around its pivot point?arrow_forwardDetermine the angular frequency of oscillation of a thin, uniform, vertical rod of mass m and length L pivoted at the point O and connected to two springs (Fig. P16.78). The combined spring constant of the springs is k(k = k1 + k2), and the masses of the springs are negligible. Use the small-angle approximation (sin ). FIGURE P16.78arrow_forward
- A very light rigid rod of length 0.500 m extends straight out from one end of a meter-stick. The combination is suspended from a pivot at the upper end of the rod as shown in Figure P12.31. The combination is then pulled out by a small angle and released. (a) Determine the period of oscillation of the system. (b) By what percentage does the period differ from the period of a simple pendulum 1.00 m long? Figure P12.31arrow_forwardReview. A system consists of a spring with force constant k = 1 250 N/m, length L = 1.50 m, and an object of mass m = 5.00 kg attached to the end (Fig. P15.49). The object is placed at the level of the point of attachment with the spring unstretched, at position yi = L, and then it is released so that it swings like a pendulum. (a) Find the y position of the object at the lowest point. (b) Will the pendulums period be greater or less than the period of a simple pendulum with the same mass m and length L? Explain. Figure PI 5.49arrow_forwardAn object of mass m1 = 9.00 kg is in equilibrium when connected to a light spring of constant k = 100 N/m that is fastened to a wall as shown in Figure P12.67a. A second object, m2 = 7.00 kg, is slowly pushed up against m1, compressing the spring by the amount A = 0.200 m (see Fig. P12.67b). The system is then released, and both objects start moving to the right on the frictionless surface. (a) When m1 reaches the equilibrium point, m2 loses contact with m1 (see Fig. P12.67c) and moves to the right with speed v. Determine the value of v. (b) How far apart are the objects when the spring is fully stretched for the first time (the distance D in Fig. P12.67d)? Figure P12.67arrow_forward
- Which of the following statements is not true regarding a massspring system that moves with simple harmonic motion in the absence of friction? (a) The total energy of the system remains constant. (b) The energy of the system is continually transformed between kinetic and potential energy. (c) The total energy of the system is proportional to the square of the amplitude. (d) The potential energy stored in the system is greatest when the mass passes through the equilibrium position. (e) The velocity of the oscillating mass has its maximum value when the mass passes through the equilibrium position.arrow_forwardWe do not need the analogy in Equation 16.30 to write expressions for the translational displacement of a pendulum bob along the circular arc s(t), translational speed v(t), and translational acceleration a(t). Show that they are given by s(t) = smax cos (smpt + ) v(t) = vmax sin (smpt + ) a(t) = amax cos(smpt + ) respectively, where smax = max with being the length of the pendulum, vmax = smax smp, and amax = smax smp2.arrow_forwardA block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a frictionless, horizontal track (Fig. P12.69). The force constant of the spring is k, and the equilibrium length is . Assume all portions of the spring oscillate in phase and the velocity of a segment of the spring of length dx is proportional to the distance x from the fixed end; that is, vx = (x/) v. Also, notice that the mass of a segment of the spring is dm = (m/) dx. Find (a) the kinetic energy of the system when the block has a speed v and (b) the period of oscillation. Figure P12.69arrow_forward
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