hw 1 solutions - university of rochesterbadolato/phy_123/resources_files/hw 1...university of...

University of Rochester Department of Physics and Astronomy Physics123, Spring 2011 Homework 1 - SOLUTIONS Conceptual Questions: • An astronaut orbiting around the earth wants to measure his weight. He has one spring with constant k and a timer. Can he infer his weight on Earth? Explain. Yes . The astronaut experiences an apparent weightlessness as in free fall (you can review this concept in Giancoli 6-4). The astronaut can form a harmonic oscillator attaching her body (of mass m) to the spring (of constant k). By minimizing possible damping forces, her motion can be well described by a simple harmonic motion. Measuring with the timer the period of oscillation, she has T = 2π m k → m = k T 2π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 . The weight (on Earth) can then be calculated by multiplying m and the acceleration of gravity at the Earth’s surface, g. • Is it possible to have damped oscillations when a system is at resonance? Explain. Yes . (Without damping the amplitude would increase without limit at resonance.) Problem 1.1 – Solution (a) The spring constant can be found from the mass and the frequency of oscillation. (b) The energy can be found from the maximum potential energy. Problem 1.2 – Solution Deuterium is the isotope of the element hydrogen with atoms having nuclei consisting of one proton and one neutron. M D 2 = 2 M H 2 → ω D 2 ω H 2 = kM D 2 kM H 2 = 1 2 → f D 2 = f H 2 2 = 0.919 × 10 14 Hz.

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Page 1: HW 1 Solutions - University of Rochesterbadolato/PHY_123/Resources_files/HW 1...University of Rochester Department of Physics and Astronomy Physics123, Spring 2011 Homework 1 - SOLUTIONS

University of Rochester Department of Physics and Astronomy

Physics123, Spring 2011

Homework 1 - SOLUTIONS Conceptual Questions: • An astronaut orbiting around the earth wants to measure his weight. He has one

spring with constant k and a timer. Can he infer his weight on Earth? Explain. Yes. The astronaut experiences an apparent weightlessness as in free fall (you can review this concept in Giancoli 6-4). The astronaut can form a harmonic oscillator attaching her body (of mass m) to the spring (of constant k). By minimizing possible damping forces, her motion can be well described by a simple harmonic motion. Measuring with the timer the period of oscillation, she has

T = 2π mk

→ m = k T2π

⎛⎝⎜

⎞⎠⎟2

The weight (on Earth) can then be calculated by multiplying m and the acceleration of gravity at the Earth’s surface, g.

• Is it possible to have damped oscillations when a system is at resonance? Explain. Yes. (Without damping the amplitude would increase without limit at resonance.) Problem 1.1 – Solution (a) The spring constant can be found from the mass and the frequency of oscillation.

(b) The energy can be found from the maximum potential energy.

Problem 1.2 – Solution Deuterium is the isotope of the element hydrogen with atoms having nuclei consisting of one proton and one neutron.

MD2= 2MH2

→ωD2

ωH2

=k MD2

k MH2

=12

→ fD2 =fH22= 0.919 ×1014Hz.