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A block on the end of a spring is pulled to position x = A and released from rest. In one full cycle of its motion, through what total distance does it travel? A/2 A 2A 4A Fig. 15.3 Consider a graphical representation (Fig. 15.3) of simple harmonic motion as described mathematically in Equation 15.6. When the object is at point A on the graph, what can you say about its position and velocity? The position and velocity are both positive. The position and velocity are both negative. The position is positive, and its velocity is zero. The position is negative, and its velocity is zero. The position is positive, and its velocity is negative. The position is negative, and its velocity is positive. Fig. 15.4 Figure 15.4 shows two curves representing objects undergoing simple harmonic motion. The correct description of these two motions is that the simple harmonic motion of object B is: of larger angular frequency and larger amplitude than that of object A. of larger angular frequency and smaller amplitude than that of object A. of smaller angular frequency and larger amplitude than that of object A. of smaller angular frequency and smaller amplitude than that of object A. An object of mass m is hung from a spring and set into oscillation. The period of the oscillation is measured and recorded as T. The object of mass m is removed and replaced with an object of mass 2m. When this object is set into oscillation, what is the period of the motion? 2T x T x T/2 Figure 15.15 shows the position of an object in uniform circular motion at t = 0. A light shines from above and projects a shadow of the object on a screen below the circular motion. What are the correct values for the amplitude and phase constant (relative to an x axis to the right) of the simple harmonic motion of the shadow? 0.50 m and 0 1.00 m and 0 0.50 m and ? 1.00 m and ? Figure 15.15 A grandfather clock depends on the period of a pendulum to keep correct time. Suppose a grandfather clock is calibrated correctly and then a mischievous child slides the bob of the pendulum downward on the oscillating rod. The grandfather clock will run: slow. fast. correctly. A grandfather clock depends on the period of a pendulum to keep correct time. Suppose a grandfather clock is calibrated correctly at sea level and is then taken to the top of a very tall mountain. The grandfather clock will now run: slow. fast. correctly. –k2x1 –k2x2 –(k1x1 + k2x2) . . The mass in the figure below slides on a frictionless surface. When the mass is pulled out, spring 1 is stretched a distance x1 from its equilibrium position and spring 2 is stretched a distance x2. The spring constants are k1 and k2 respectively. The force pulling back on the mass is: A body of mass 5.0 kg is suspended by a spring which stretches 10 cm when the mass is attached. It is then displaced downward an additional 5.0 cm and released. Its position as a function of time is approximately y = .10 sin 9.9t y = .10 cos 9.9t y = .10 cos (9.9t + .1) y = .10 sin (9.9t + 5) y = .05 cos 9.9t A body oscillates with simple harmonic motion along the x-axis. Its displacement varies with time according to the equation x = 5.0 sin (pt). The acceleration (in m/s2) of the body at t = 1.0 s is approximately 3.5 49 14 43 4.3 A body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = 5 sin (pt + p/3). The phase (in rad) of the motion at t = 2 s is 7p/3 p/3 p 5p /3 2p A body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = 5.0 sin (pt + p/3). The velocity (in m/s) of the body at t = 1.0 s is +8.0 –8.0 –14 +14 –5.0 A mass m = 2.0 kg is attached to a spring having a force constant k = 290 N/m as in the figure. The mass is displaced from its equilibrium position and released. Its frequency of oscillation (in Hz) is approximately 12 0.50 0.01 1.9 0.08 The mass in the figure slides on a frictionless surface. If m = 2 kg, k1 = 800 N/m and k2 = 500 N/m, the frequency of oscillation (in Hz) is approximately 6 2 4 8 10 A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. A point or points at which the object has positive velocity and zero acceleration is(are) B C D B or D A or E A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. The point at which the object has negative velocity and zero acceleration is A B C D E A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. The point at which the object has zero velocity and positive acceleration is A B C D E A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below. The point at which the object has zero velocity and negative acceleration is A B C D E In an inertia balance, a body supported against gravity executes simple harmonic oscillations in a horizontal plane under the action of a set of springs. If a 1.00 kg body vibrates at 1.00 Hz, a 2.00 kg body will vibrate at 0.500 Hz. 0.707 Hz. 1.00 Hz. 1.41 Hz. 2.00 Hz. Suppose it were possible to drill a frictionless cylindrical channel along a diameter of the Earth from one side of the Earth to another. A body dropped into such a channel will only feel the gravitational pull of mass within a sphere of radius equal to the distance of the mass from the center of the Earth . The density of the Earth is and . The mass will oscillate with a period of 84.4 min. 169 min. 24.0 h. 1130 h. 27.2 d. Ellen says that whenever the acceleration is directly proportional to the displacement of an object from its equilibrium position, the motion of the object is simple harmonic motion. Mary says this is true only if the acceleration is opposite in direction to the displacement. Which one, if either, is correct? Ellen, because w2 is directly proportional to the constant multiplying the displacement and to the mass. Ellen, because w2 is directly proportional to the mass. Mary, because w2 is directly proportional to the constant multiplying the displacement and to the mass. Mary, because w2 is directly proportional to the mass. Mary, because the second derivative of an oscillatory function like sin(wt) or cos(wt) always is the negative of the original function. John says that the value of the function cos[w(t + T) + j] , obtained one period T after time t, is greater than cos(wt + j) by 2p. Larry says that it is greater by the addition of 1.00 to cos(wt + j). Which one, if either, is correct? John, because wT = 2p. John, because wT = 1 radian. Larry, because wT = 2p. Larry, because wT = 1 radian. Neither, because cos(q + 2p) = cosq. The motion of a particle connected to a spring is described by x = 10 sin (pt). At what time (in s) is the potential energy equal to the kinetic energy? 0 0.25 0.50 0.79 1.0 The amplitude of a system moving with simple harmonic motion is doubled. The total energy will then be 4 times larger 3 times larger 2 times larger the same as it was half as much To double the total energy of a mass oscillating at the end of a spring with amplitude A, we need to increase the angular frequency by . increase the amplitude by . increase the amplitude by 2. increase the angular frequency by 2. increase the amplitude by 4 and decrease the angular frequency by . Simple harmonic oscillations can be modeled by the projection of circular motion at constant angular velocity onto a diameter of the circle. When this is done, the analog along the diameter of the centripetal acceleration of the particle executing circular motion is the displacement from the center of the diameter of the projection of the position of the particle on the circle. the projection along the diameter of the velocity of the particle on the circle. the projection along the diameter of tangential acceleration of the particle on the circle. the projection along the diameter of centripetal acceleration of the particle on the circle. meaningful only when the particle moving in the circle also has a non-zero tangential acceleration. Two circus clowns (each having a mass of 50 kg) swing on two flying trapezes (negligible mass, length 25 m) shown in the figure. At the peak of the swing, one grabs the other, and the two swing back to one platform. The time for the forward and return motion is 10 s 50 s 15 s 20 s 25 s w3 > w2 > w1 Need to know amplitudes to answer this question. Need to know to answer this question. w1 > w2 > w3 w1 = w2 = w3 Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles q1, q2 and q3 respectively and released. The approximation sin q = q holds for all three angles, with q3 > q2 > q1. How do the angular frequencies of the three pendulums compare? When a damping force is applied to a simple harmonic oscillator which has angular frequency w0 in the absence of damping, the new angular frequency w is such that w < w0. w = w0. w > w0. wT < w0T0. wT > w0T0. When a damping force is applied to a simple harmonic oscillator which has period T0 in the absence of damping, the new period T is such that T < T0. T = T0. T > T0. wT < w0T0. wT > w0T0.
