Question

A heavy rope 6.00 m long and weighing 29.4 N is attached at one end to a ceiling and hangs vertically. A 0.500-kg mass is suspended from the lower end of the rope. What is the speed of transverse waves on the rope at the

(a) bottom of the rope, (b) middle of the rope, and (c) top of the rope? (d) Is the tension in the middle of the rope the average of the tensions at the top and bottom of the rope? Is the wave speed at the middle of the rope the average of the wave speeds at the top and bottom? Explain.

Answer 1

Answer:

a) $v=3.1252\ m.s^{-1}$

b) $v=39.0672\ m.s^{-1}$

c) $v=8.2685\ m.s^{-1}$

d) No,

   No.

Explanation:

Given:

length of rope, $l=6\ m$

weight of the rope, $w=29.4\ N$

mass suspended at the lower end of the rope, $M=0.5\ kg$

Now the mass of the rope:

$m=\frac{w}{g}$

$m=\frac{29.4}{9.8}$

$m=3.01\ kg$

So the linear mass density of rope:

$\mu=\frac{m}{l}$

$\mu=\frac{3.01}{6}$

$\mu=0.5017\ kg.m^{-1}$

We know that the speed of wave in a tensed rope is given as:

$v=\sqrt{\frac{F_T}{\mu} }$

where:

$F_T=$  tension force in the rope

a)

At the bottom of the hanging rope we have an extra mass suspended. So the tension at the bottom of the rope:

$F_T=M\times g$

$F_T=0.5\times 9.8$

$F_T=4.9\ N$

Therefore the speed of the wave at the bottom point of the rope:

$v=\sqrt{\frac{4.9}{0.5017} }$

$v=3.1252\ m.s^{-1}$

b)

Tension at a point in the middle of the rope:

$F_T=M\times g+\frac{w}{2}$

$F_T=0.5\times 9.8+\frac{29.4}{2}$

$F_T=19.6\ N$

Now wave speed at this point:

$v=\sqrt{\frac{19.6}{0.5017} }$

$v=39.0672\ m.s^{-1}$

c)

Tension at a point in the top of the rope:

$F_T=M\times g+w$

$F_T=0.5\times 9.8+29.4$

$F_T=34.3\ N$

Now wave speed at this point:

$v=\sqrt{\frac{34.3}{0.5017} }$

$v=8.2685\ m.s^{-1}$

d)

Tension at the middle of the rope is not the average tension of tension at the top and bottom of the rope because we have an extra mass attached at the bottom end of the rope.

Also the wave speed at the mid of the rope is not the average f the speeds at the top and the bottom of the ropes because it depends upon the tension of the rope at the concerned points.

Answer 2

The wave speed at a given section of the rope directly depends on the

square root of the tension that is present at the section.

The correct responses are;

• (a) 3.143 m/s

• (b) 6.256 m/s

• (c) 8.29 m/s

• (d) Yes

• (d second part) No

Reasons:

Length of the rope, L = 6.00 m

Weight of the rope, W = 29.4 N

Mass at the lower end of the rope, m = 0.500-kg

(a) Required:

The speed, $v_b$, of the traverse wave at the bottom of the rope.

Solution:

$\displaystyle v = \mathbf{\sqrt{\frac{T}{\mu} }}$

Where;

T = The tension force in the specified part of the rope

At the bottom of the rope, the tension is due to weight of the mass m only

Therefore;

At the bottom of the rope, $\mathbf{T_b}$ = m × g = Weight of the mass at the lower end

g = Acceleration due to gravity

Weight of the mass = $T_b$ = 0.500 kg × 9.81 m/s² = 4.905 N

$\displaystyle \mu = Mass \ per \ unit \ length = \frac{W}{g \cdot L} = \mathbf{ \frac{29.4 \, N}{9.81 \, m/s^2 \times 6.00 \, m}}$

Therefore;

$\displaystyle v_b = \sqrt{\frac{T_b}{\frac{W}{g \cdot L} } } = \sqrt{\frac{4.905}{\frac{29.4}{9.81 \times 6} } } \approx \mathbf{3.134}$

The speed of the transverse wave at the bottom of the rope, $v_b$ ≈ 3.134 m/s

(b) Required:

The speed of the transverse wave at the middle of the rope, $v_m$

Solution:

At the middle of the rope, the value of $T_m$, the tension in the rope is given as follows;

$T_m$ = Weight of mass at the lower end + Weight of half the rope

Therefore;

$T_m$ = 4.905 N + 0.5 × 29.4 N = 19.605 N

Which gives;

• $\displaystyle v_m = \sqrt{\frac{19.605}{\frac{29.4}{9.81 \times 6} } } \approx 6.265$

The speed of the transverse wave at the middle of the rope, $v_m$ ≈ 6.265 m/s

(c) Required:

The speed of the transverse wave at the top of the rope, $\mathbf{v_T}$

Solution:

At the top of the rope, T = Weight of the rope + Weight of the mass

Therefore;

$T_T$ = 29.4 N + 4.905 N = 34.305 N

Therefore;

$\displaystyle v_T = \sqrt{\frac{34.305}{\frac{29.4}{9.81 \times 6} } } \approx \mathbf{ 8.29}$

The speed of the transverse wave at the top of the rope, $v_T$ ≈ 8.29 m/s

(d) The tension at the middle of the rope $T_m$ is found as follows;

$T_m$ = Half the weight of the rope + The weight at the lower end of the rope

Therefore;

$\displaystyle Tm = \mathbf{\frac{W}{2} + m \cdot g}$

The average tension at the top of the rope is given as follows;

$\displaystyle T_{Tave} = \mathbf{\frac{Tension \ at \ the \ top + Tension \ at \ the \ bottom}{2}}$

The tension at the top = Weight of rope + Weight of the mass

Therefore;

Tension at the top = W + m·g

Tension at the bottom = m·g

Which gives;

$\displaystyle T_{Tave} = \frac{W + m \cdot g + m \cdot g}{2} = \frac{W + 2 \cdot m \cdot g}{2} = \mathbf{m \cdot g + \frac{W}{2}}$

$\displaystyle T_{Tave} = m \cdot g + \frac{W}{2} = \frac{W}{2} + m\cdot g = T_m$

$T_{Tave} = T_m$

Therefore;

Yes

The tension in the middle of the rope is the same as the average of the tension at the top of the rope.

The wave speed at the middle of the rope, $\displaystyle v_m = \sqrt{\frac{T_m}{\frac{W}{g \cdot L} } } = \mathbf{\frac{1}{\sqrt{\frac{W}{g \cdot L} } } \times \sqrt{T_m}}$

The average wave speed at the top and bottom of the rope is given as follows;

$\displaystyle v_{(ave, \, T \, and \, b)} = \mathbf{\frac{v_{T} + v_b }{2}}$

Which gives;

$v_{(ave, \, T \, and \, b)} = \displaystyle \frac{\sqrt{\frac{T_T}{\frac{W}{g \cdot L} } } + \sqrt{\frac{T_b}{\frac{W}{g \cdot L} } }}{2} = \mathbf{ \frac{1}{\sqrt{\frac{W}{g \cdot L} } } \times \frac{1}{2} \times \left(\sqrt{T_T} + \sqrt{T_b} \right)}$

However;

$\displaystyle \frac{1}{2} \times \left(\sqrt{T_T} + \sqrt{T_b} \right) \approx 5.38$

$\sqrt{T_m} \approx 4.43$

Therefore;

$\displaystyle \frac{1}{2} \times \left(\sqrt{T_T} + \sqrt{T_b} \right) \neq \sqrt{T_m}$

$v_m \neq \mathbf{v_{(ave, \, T\, and \, b )}}$

Therefore;

No

The wave speed at the middle of the rope is not the same as the average wave speeds at the top and bottom.

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