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Chapter 13: Problem 53

Transverse waves with a speed of \(50.0 \mathrm{~m} / \mathrm{s}\) are to be produced on a stretched string. A \(5.00-\mathrm{m}\) length of string with a total mass of \(0.0600 \mathrm{~kg}\) is used. (a) What is the required tension in the string? (b) Calculate the wave speed in the string if the tension is \(8.00 \mathrm{~N}\).

Short Answer

Expert verified
The required tension to produce transverse waves with a speed of 50.0 m/s on the string is 30.0 N (Newton). If the tension in the string is 8.00 N, the wave speed is 28.9 m/s.

Step by step solution

01

Determining the mass per unit length (linear density) of the string

The linear density \( \mu \) of a string is defined as its mass per unit length. It's given by the formula \( \mu = \frac{m}{L} \), where \(m\) is the total mass of the string and \(L\) is its length. In this case, \(m = 0.0600 \ kg \) and \( L = 5.00 \ m \). Substituting these values into the formula gives \( \mu = \frac{0.0600}{5.00} = 0.012 \ kg/m \).
02

Calculating the required tension in the string

The formula that relates the speed \(v\) of a wave on a string, the tension \(T\) in the string, and the string's linear density \( \mu \) is \( v = \sqrt{\frac{T}{\mu}} \). In this problem, the wave speed \( v = 50.0 \ m/s \) and \( \mu = 0.012 \ kg/m \) (from Step 1) are given, and the tension \( T \) is what we need to find. Re-arranging the formula for \( T \) gives \( T = \mu \times v^2 \). Substituting the given values into this formula gives \( T = 0.012 \ kg/m \times (50.0 \ m/s)^2 = 30.0 \ N \). So, the required tension in the string is 30.0 N.
03

Calculating the wave speed when the tension is 8.00 N

The final part of the problem requires us to find the wave speed when the tension is given as \( T = 8.00 \ N \). The same formula for wave speed we used in Step 2 applies here: \( v = \sqrt{\frac{T}{\mu}} \). Substituting the new tension and the linear density found in Step 1 into this equation gives \( v = \sqrt{\frac{8.00 \ N}{0.012 \ kg/m}} = 28.9 \ m/s \). So the wave speed when the tension is 8.00 N is 28.9 m/s.

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Most popular questions from this chapter

A \(0.250-\mathrm{kg}\) block resting on a frictionless, horizontal surface is attached to a spring having force constant \(\quad 83.8 \mathrm{~N} / \mathrm{m}\) as in Figure P13.16. A horizontal force \(\overrightarrow{\mathbf{F}}\) causes the spring to stretch a distance of \(5.46 \mathrm{~cm}\) from its equilibrium position. (a) Find the value of \(F\). (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block immediately after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, how would your answer to part (d) change? (f) What other information would you need to know to find the actual answer to part (d) in this case?

The position of a \(0.30-\mathrm{kg}\) object attached to a spring is described by $$ x=(0.25 \mathrm{~m}) \cos (0.4 \pi t) $$ Find (a) the amplitude of the motion, (b) the spring constant, (c) the position of the object at \(t=0.30 \mathrm{~s}\), and (d) the object's speed at \(t=0.30 \mathrm{~s}\).

An object moves uniformly around a circular path of radius \(20.0 \mathrm{~cm}\), making one complete revolution every \(2.00 \mathrm{~s}\). What are (a) the translational speed of the object, (b) the frequency of motion in hertz, and (c) the angular speed of the object?

A horizontal object-spring system oscillates with an amplitude of \(3.5 \mathrm{~cm}\) on a frictionless surface. If the spring constant is \(250 \mathrm{~N} / \mathrm{m}\) and the object has a mass of \(0.50 \mathrm{~kg}\), determine (a) the mechanical energy of the system, (b) the maximum speed of the object, and (c) the maximum acceleration of the object.

A spring of negligible mass stretches \(3.00 \mathrm{~cm}\) from its relaxed length when a force of \(7.50 \mathrm{~N}\) is applied. A \(0.500-\mathrm{kg}\) particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to \(x=5.00 \mathrm{~cm}\) and released from rest at \(t=0\). (a) What is the force constant of the spring? (b) What are the angular frequency \(\omega\), the frequency, and the period of the motion? (c) What is the total energy of the system? (d) What is the amplitude of the motion? (e) What are the maximum velocity and the maximum acceleration of the particle? (f) Determine the displacement \(x\) of the particle from the equilibrium position at \(t=0.500 \mathrm{~s}\). (g) Determine the velocity and acceleration of the particle when \(t=0.500 \mathrm{~s}\).
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