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# # $ ' # (a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as x = 0. The object is pulled down an additional 18.0 cm and released from rest to oscillate without friction. What is its position x at a time 84.4 s later? # . # . / .# ( !0 /
. (! / ' /1 (2 2 (2 /12 (/2 (/ 2 6. The initial position, velocity, and acceleration of an object moving in simple harmonic motion are xi, vi, and a i; the angularfrequency of oscillation is ω. (a) Show that the position and velocity of the object for all time can be written as x(t) = x i cos ωt + vi sin ωt ω v(t) = – xi ω sin ω t + vi cos ω t (b) If the amplitude of the motion is A, show that v2 – ax = vi2 – a i xi = ω 2A2 " # 3 3' , , , This is the answer to (a). 456 10.A piston in a gasoline engine is in simple harmonic motion. If the extremes of its position relative to its center point are ± 5.00 cm, find the maximum velocity and acceleration of the piston when the engine is running at the rate of 3 600 rev/min. ! ! 77 / 28 (2 2 7( 7 / 13.A 1.00-kg object is attached to a horizontal spring. The spring is initially stretched by 0.100 m, and the object is released from rest there. It proceeds to move without friction. The next time the speed of the object is zero is 0.500 s later. What is the maximum speed of the object? We do not know the angular velocity, but the motion is described by ( The first time its speed is zero after release is when the spring is fully compressed, which will occur after half a cycle, or / !2 (!2 !2 16. A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 2.00 J, find (a) the force constant of the spring and (b) the amplitude of the motion. Energy of an oscillator is The angular frequency is
/( . . !( (! ( !0 (72 ( . 9 ( The amplitude of a system moving in simple harmonic motion is doubled. Determine the change in (a) the total energy, (b) the maximum speed, (c) the maximum acceleration, and (d) the period. (a) Total energy is proportional to A^2, so it will quadruple (b) Maximum speed is proportional to A, so it will double )* ) *: $ ' 3 ' 7 A man enters a tall tower, needing to know its height. He notes that a long pendulum extends from the ceiling almost to the floor and that its period is 12.0 s. (a) How tall is the tower? (b) What If? If this pendulumis taken to the Moon, where the free-fall 2 acceleration is 1.67 m/s , what is its period there? (a) 9 (b)
9 !2 /7 ( !7 1 # 9 # /7 , 30. The angular position of a pendulumis represented by the equation θ = (0.320 rad) cos ω t, where θ is in radians and ω = 4.43 rad/s. Determine the period and length of the pendulum.
9 9 9 # !2 ( (7 # 12 38.A torsional pendulumis formed by taking a meter stick of mass 2.00 kg, and attaching to its center a wire. With its upper end clamped, the vertical wire supports the stick as the stick turns in a horizontal plane. If the resulting period is 3.00 minutes, what is the torsion constant for the wire? We first need to calculate the moment of inertia of the stick about its center of mass. + (
+ .# ( ( ( .# ! Now we use Hence 9 + 9 ! /2 (2 ( .# " &' ) #* 66. A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a horizontal, frictionless track (Fig. P15.66). The force constant of the spring is k and the equilibrium length is . Assume that all portions of the spring oscillate in phase and that the velocity of a segment dx is proportional to the distance x from the fixed end; that is, vx = (x/ )v. Also, note 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. Kinetic energy of spring and mass when the speed of mass M is v is
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