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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

An \(80.0-\mathrm{kg}\) bungee jumper is enjoying an afternoon of jumps. The jumper's first oscillation has an amplitude of \(10.0 \mathrm{~m}\) and a period of \(5.00 \mathrm{~s}\). Treating the bungee cord as spring with no damping, calculate each of the following: a) the spring constant of the bungee cord. b) the bungee jumper's maximum speed during the ascillation, and c) the time for the amplitude to decrease to \(2.00 \mathrm{~m}\) (with air resistance providing the damping of the oscillations at \(7.50 \mathrm{~kg} / \mathrm{s})\)

A horizontal tree branch is directly above another horizontal tree branch. The elevation of the higher branch is \(9.65 \mathrm{~m}\) above the ground, and the elevation of the lower branch is \(5.99 \mathrm{~m}\) above the ground. Some children decide to use the two branches to hold a tire swing. One end of the tire swing's rope is tied to the higher tree branch so that the bottom of the tire swing is \(0.47 \mathrm{~m}\) above the ground. This swing is thus a restricted pendulum. Start. ing with the complete length of the rope at an initial angle of \(14.2^{\circ}\) with respect to the vertical, how long does it take a child of mass \(29.9 \mathrm{~kg}\) to complete one swing back and forth?

Object \(A\) is four times heavier than object B. Fach object is attached to a spring, and the springs have equal spring constants. The two objects are then pulled from their equilibrium positions and released from rest. What is the ratio of the periods of the two oscillators if the amplitude of \(A\) is half that of \(B ?\) a) \(T_{A}: T_{\mathrm{g}}=1: 4\) c) \(T_{A}: T_{\mathrm{B}}=2\) : b) \(T_{A}: T_{B}=4: 1\) d) \(T_{A}: T_{B}=1: 2\)

Pendulum A has a bob of mass \(m\) hung from a string of length \(I_{i}\) pendulum \(B\) is identical to \(A\) except its bob has mass \(2 m\). Compare the frequencies of small oscillations of the two pendulums.

A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass \(M\) and length \(L\) that is pivoted freely about one end. with a solid sphere of the same mass, \(M,\) and a radius of \(L / 2\) centered about the free end of the rod. a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of \(M\) and \(L\). b) Obtain an expression for the period of the pendulum for small oscillations.
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