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Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 1 [20 pts] The position of a particle is given by x = (7 cm) cos(6πt), where t is in seconds. What are (a) the frequency, (b) the period, and (c) the amplitude of the particle's motion? (d) What is the first time after t = 0 that the particle is at its equilibrium position? In what direction is it moving at that time? (e) Sketch x as a function of t. (f) Find the distance traveled in the first, second, third, and fourth second after t = 0. Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 2 [20 pts] Figure shows a uniform disk of radius R = 0.8 m and 6-kg mass with a small hole a distance d from the disk's center that can serve as a pivot point. (a) What should be the distance d so that the period of this physical pendulum is 2.5 s? (b) What should be the distance d so that this physical pendulum will have the shortest possible period? What is this period?
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 3 [20 pts] A simple pendulum of length L is attached to a massive cart that slides without friction down a plane inclined at angle θ with the horizontal as shown in Figure. Find the period of oscillation of the pendulum on the sliding cart.
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 4 [20 pts] Figure shows a pendulum of length L with a bob of mass M. The bob is attached to a spring of spring constant k as shown. When the bob is directly below the pendulum support, the spring is at its equilibrium length. (a) Derive an expression for the period of this oscillating system for small amplitude vibrations. Please, derive first using the dynamical method, then the energy method. Show that they give the same result. (b) Suppose that M = 1 kg and L is such that in the absence of the spring the period is 2.0 s. What is the spring constant k if the period of the oscillating system is 1.0 s?
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 5 [20 pts] Find the resonance frequency for each of the three systems shown in figure.
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Optional Problems Please note that you do not have to do the optional problems. However, I expect ambitious students to do these problems, too. Optional Problem 1: Figure shows a uniform solid half-cylinder of mass M and radius R resting on a horizontal surface. If one side of this cylinder is pushed down slightly and then released, the object will oscillate about its equilibrium position. Determine the period of this oscillation. .
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Optional Problem 2: The Morse potential, which is often used to model interatomic forces, can be written in the form U (r ) = D(1 - e - b ( r - r0 ) ) 2 , where r is the distance between the two
atomic nuclei. (a) Using a spreadsheet program or graphing calculator, make a graph of the Morse potential using D = 5 eV, β= 0.2 nm- 1 , and r0 = 0 . 75 nm. (b) Determine the equilibrium separation and "spring constant" for small displacements from equilibrium for the Morse potential. (c) Determine a formula for the oscillation frequency for a homonuclear diatomic molecule (that is, two of the same atoms), where the atoms have mass m .
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 1 [20 pts] The position of a particle is given by x = (7 cm) cos(6πt), where t is in seconds. What are (a) the frequency, (b) the period, and (c) the amplitude of the particle's motion? (d) What is the first time after t = 0 that the particle is at its equilibrium position? In what direction is it moving at that time? (e) Sketch x as a function of t. (f) Find the distance traveled in the first, second, third, and fourth second after t = 0. Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 2 [20 pts] Figure shows a uniform disk of radius R = 0.8 m and 6-kg mass with a small hole a distance d from the disk's center that can serve as a pivot point. (a) What should be the distance d so that the period of this physical pendulum is 2.5 s? (b) What should be the distance d so that this physical pendulum will have the shortest possible period? What is this period?
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 3 [20 pts] A simple pendulum of length L is attached to a massive cart that slides without friction down a plane inclined at angle θ with the horizontal as shown in Figure. Find the period of oscillation of the pendulum on the sliding cart.
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 4 [20 pts] Figure shows a pendulum of length L with a bob of mass M. The bob is attached to a spring of spring constant k as shown. When the bob is directly below the pendulum support, the spring is at its equilibrium length. (a) Derive an expression for the period of this oscillating system for small amplitude vibrations. Please, derive first using the dynamical method, then the energy method. Show that they give the same result. (b) Suppose that M = 1 kg and L is such that in the absence of the spring the period is 2.0 s. What is the spring constant k if the period of the oscillating system is 1.0 s?
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Problem 5 [20 pts] Find the resonance frequency for each of the three systems shown in figure.
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Optional Problems Please note that you do not have to do the optional problems. However, I expect ambitious students to do these problems, too. Optional Problem 1: Figure shows a uniform solid half-cylinder of mass M and radius R resting on a horizontal surface. If one side of this cylinder is pushed down slightly and then released, the object will oscillate about its equilibrium position. Determine the period of this oscillation. .
Solution:
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
Optional Problem 2: The Morse potential, which is often used to model interatomic forces, can be written in the form U (r ) = D(1 - e - b ( r - r0 ) ) 2 , where r is the distance between the two
atomic nuclei. (a) Using a spreadsheet program or graphing calculator, make a graph of the Morse potential using D = 5 eV, β= 0.2 nm- 1 , and r0 = 0 . 75 nm. (b) Determine the equilibrium separation and "spring constant" for small displacements from equilibrium for the Morse potential. (c) Determine a formula for the oscillation frequency for a homonuclear diatomic molecule (that is, two of the same atoms), where the atoms have mass m .
Chapter 15
[HOMEWORK 4]
Return by Wednesday, November 10
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