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Chapter 14: Problem 18

You have a linear (following Hooke's Law) spring with an unknown spring constant, a standard mass, and a timer. Explain carefully how you could most practically use these to measure masses in the absence of gravity. Be as quantitative as you can. Regard the mass of the spring as negligible

Short Answer

Expert verified
Answer: You can measure masses in the absence of gravity by first determining the spring constant (k) using Hooke's Law and the standard mass. Then, set up the mass-spring system in zero gravity, attach the unknown mass to the spring and oscillate it, measuring the period of oscillation (T). Next, calculate the angular frequency (ω) and use it, along with the spring constant (k), to solve for the unknown mass (M).

Step by step solution

01

Determine the spring constant (k) using Hooke's Law

Place the standard mass on the spring and measure the displacement (d) caused by the attached standard mass. According to Hooke's Law, F = -k * d, where F is the force exerted by the spring, k is the spring constant, and d is the displacement. In this case, F is the gravitational force acting on the standard mass (F = m * g), where m is the mass, and g is the acceleration due to gravity. Therefore, we can find the spring constant k by rearranging the equation: k = -(m * g) / d.
02

Set up the mass-spring system in zero gravity

For the given problem, we have to operate in the absence of gravity. Set up the mass-spring system in such a way that gravity is no longer a factor, meaning the spring will only provide the restoring force for any attached mass.
03

Oscillate the spring and measure the period

Attach the unknown mass (M) to the spring and slightly displace it from its equilibrium position to set it into oscillation. Using the timer, measure the time it takes for the mass to complete one full oscillation. This time is known as the period of oscillation (T).
04

Calculate the angular frequency

Since we have the period of oscillation, we can calculate the angular frequency of oscillations (ω) using the following equation: ω = (2 * π) / T.
05

Calculate the unknown mass (M) using the angular frequency

The angular frequency (ω) is related to the spring constant (k) and the unknown mass (M) with the following equation: ω² = k / M. Use the previously calculated values of k and ω to solve for the unknown mass (M): M = k / ω². By following these steps, one could practically use a linear spring following Hooke's Law, a standard mass, and a timer to measure masses in the absence of gravity.

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Most popular questions from this chapter

The motion of a planet in a circular orbit about a star obeys the equations of simple harmonic motion. If the orbit is observed edge-on, so the planet's motion appears to be onedimensional, the analogy is quite direct: The motion of the planet looks just like the motion of an object on a spring a) Use Kepler's Third Law of planetary motion to determine the "spring constant" for a planet in circular orbit around a star with period \(T\) b) When the planet is at the extremes of its motion observed edge-on, the analogous "spring" is extended to its largest displacement. Using the "spring" analogy, determine the orbital velocity of the planet.

A 2.0 -kg mass attached to a spring is displaced \(8.0 \mathrm{~cm}\) from the equilibrium position. It is released and then oscillates with a frequency of \(4.0 \mathrm{~Hz}\) a) What is the energy of the motion when the mass passes through the equilibrium position? b) What is the speed of the mass when it is \(20 \mathrm{~cm}\) from the equilibrium position?

The period of a pendulum is \(0.24 \mathrm{~s}\) on Earth. The period of the same pendulum is found to be 0.48 s on planet \(X,\) whose mass is equal to that of Earth. (a) Calculate the gravitational acceleration at the surface of planet \(X\). (b) Find the radius of planet \(\mathrm{X}\) in terms of that of Earth.

An object in simple harmonic motion is isochronous, meaning that the period of its oscillations is independent of their amplitude. (Contrary to a common assertion, the operation of a pendulum clock is not based on this principle. A pendulum clock operates at fixed, finite amplitude. The gearing of the clock compensates for the anharmonicity of the pendulum.) Consider an oscillator of mass \(m\) in one-dimensional motion, with a restoring force \(F(x)=-c x^{3}\) where \(x\) is the displacement from equilibrium and \(c\) is a constant with appropriate units. The motion of this ascillator is periodic but not isochronous. a) Write an expression for the period of the undamped oscillations of this oscillator. If your expression involves an integral, it should be a definite integral. You do not need to evaluate the expression. b) Using the expression of part (a), determine the dependence of the period of oscillation on the amplitude. c) Generalize the results of parts (a) and (b) to an oscillator of mass \(m\) in one-dimensional motion with a restoring force corresponding to the potential energy \(U(x)=\gamma|x| / \alpha\), where \(\alpha\) is any positive value and \(\gamma\) is a constant

A small mass, \(m=50.0 \mathrm{~g}\), is attached to the end of a massless rod that is hanging from the ceiling and is free to swing. The rod has length \(L=1.00 \mathrm{~m} .\) The rod is displaced \(10.0^{\circ}\) from the vertical and released at time \(t=0\). Neglect air resistance. What is the period of the rod's oscillation? Now suppose the entire system is immersed in a fluid with a small damping constant, \(b=0.0100 \mathrm{~kg} / \mathrm{s},\) and the rod is again released from an initial displacement angle of \(10.0^{\circ}\). What is the time for the amplitude of the oscillation to reduce to \(5.00^{\circ}\) ? Assume that the damping is small Also note that since the amplitude of the oscillation is small and all the mass of the pendulum is at the end of the rod, the motion of the mass can be treated as strictly linear, and you can use the substitution \(R \theta(t)=x(t),\) where \(R=1.0 \mathrm{~m}\) is the length of the pendulum rod.
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