Showing 1 to 20 of 2092 Questions

• A ball dropped from a height of 4.00 m makes a perfectly elastic collision with the ground. Assuming no mechanical energy is lost due to air resistance, (a) Show that the ensuing motion is periodic and (b) Determine the period of the motion. (c) Is the motion simple harmonic? Explain.
• In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x = (5.00 cm) cos (2t + /6) where x is in centimeters and t is in seconds. At t = 0, find
(a) The position of the piston,
(b) Its velocity, and
(c) Its acceleration.
(d) Find the period and amplitude of the motion.
• The position of a particle is given by the expression x = (4.00 m) cos (3.00)t % )), where x is in meters and t is in seconds. Determine
(a) The frequency and period of the motion,
(b) The amplitude of the motion,
(c) The phase constant, and
(d) The position of the particle at t = 0.250 s.
• (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?
(b) What If? A hanging spring stretches by 35.5 cm when an object of mass 440 g is hung on it at rest. We define this new position as x = 0. This object is also pulled down an additional 18.0 cm and released from rest to oscillate without friction. Find its position 84.4 s later.
(c) Why are the answers to (a) and (b) different by such a large percentage when the data are so similar? Does this circumstance reveal a fundamental difficulty in calculating the future?
(d) Find the distance traveled by the vibrating object in part (a).
(e) Find the distance traveled by the object in part (b).
• A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz.
(a) Show that the position of the particle is given by x = (2.00cm) sin (3.00 t) Determine
(b) The maximum speed and the earliest time (t > 0) at which the particle has this speed,
(c) The maximum acceleration and the earliest time (t > 0) at which the particle has this acceleration, and
(d) The total distance traveled between t = 0 and t = 1.00 s.
• The initial position, velocity, and acceleration of an object moving in simple harmonic motion are xi, vi, and ai; the angular frequency of oscillation is w.
(a) Show that the position and velocity of the object for all time can be written as
x (t) = x i cos wt + (vi/w) sin wt
v (t) = ─ xiw sin wt + vi cos wt
(b) If the amplitude of the motion is A, show that v2 ─ ax = vi2 ─ aixi = w2A2
• A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find
(a) The period of its motion,
(b) The frequency in hertz, and
(c) The angular frequency in radians per second.
• A vibration sensor, used in testing a washing machine, consists of a cube of aluminum 1.50 cm on edge mounted on one end of a strip of spring steel (like a hacksaw blade) that lies in a vertical plane. The mass of the strip is small compared to that of the cube, but the length of the strip is large compared to the size of the cube. The other end of the strip is clamped to the frame of the washing machine, which is not operating. A horizontal force of 1.43 N applied to the cube is required to hold it 2.75 cm away from its equilibrium position. If the cube is released, what is its frequency of vibration?
• A 7.00-kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a period of 2.60 s. Find the force constant of the spring.
• 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.
• A 0.500-kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion with an amplitude of 10.0 cm. Calculate
(a) The maximum value of its speed and acceleration,
(b) The speed and acceleration when the object is 6.00 cm from the equilibrium position, and
(c) The time interval required for the object to move from x = 0 to x = 8.00 cm.
• A 1.00-kg glider attached to a spring with a force constant of 25.0 N/m oscillates on a horizontal, frictionless air track. At t = 0 the glider is released from rest at x = ─3.00 cm. (That is, the spring is compressed by 3.00 cm.) Find
(a) The period of its motion,
(b) The maximum values of its speed and acceleration, and
(c) The position, velocity, and acceleration as functions of time.
• 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?
• A particle that hangs from a spring oscillates with an angular frequency '. The spring is suspended from the ceiling of an elevator car and hangs motionless (relative to the elevator car) as the car descends at a constant speed v. The car then stops suddenly.
(a) With what amplitude does the particle oscillate?
(b) What is the equation of motion for the particle? (Choose the upward direction to be positive.)
• A block of unknown mass is attached to a spring with a spring constant of 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm. When the block is halfway between its equilibrium position and the end point, its speed is measured to be 30.0 cm/s. Calculate (a) the mass of the block,
(b) the period of the motion, and (c) the maximum acceleration of the block.
• 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.
• An automobile having a mass of 1 000 kg is driven into a brick wall in a safety test. The bumper behaves like a spring of force constant 5.00, 106 N/m and compresses 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming that no mechanical energy is lost during impact with the wall?
• A block–spring system oscillates with an amplitude of 3.50 cm. If the spring constant is 250 N/m and the mass of the block is 0.500 kg, determine
(a) The mechanical energy of the system,
(b) The maximum speed of the block, and
(c) The maximum acceleration.
• A 50.0-g object connected to a spring with a force constant of 35.0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4.00 cm. Find
(a) The total energy of the system and
(b) The speed of the object when the position is 1.00 cm. Find
(c) The kinetic energy and
(d) The potential energy when the position is 3.00 cm.
• A 2.00-kg object is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest with an initial position of xi = 0.200 m, and it subsequently undergoes simple harmonic oscillations. Find
(a) The force constant of the spring,
(b) The frequency of the oscillations, and
(c) The maximum speed of the object. Where does this maximum speed occur?
(d) Find the maximum acceleration of the object. Where does it occur?
(e) Find the total energy of the oscillating system. Find
(f) The speed and
(g) The acceleration of the object when its position is equal to one third of the maximum value.