Any help would be appreciated. A heavy rope 6.00 m long and weighing 20.4 N is attached at one end to a ceiling and hangs vertically. A 0.510-kg mass is suspended from the lower end of the rope.

asked by @eliannav1 • almost 2 years ago • Physics • 5 pts

Part A- What is the speed of transverse waves on the rope at the bottom of the rope? Part B- What is the speed of transverse waves on the rope at the middle of the rope? Part C- What is the speed of transverse waves on the rope at the top of the rope? Part 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? Select the correct answer and explanation.

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1 answer

First, this question deals only with the speed of transverse waves on a string, so we need the appropriate equation:

v = sqrt(T/mu)

where T is the tension in the string and mu is the mass per unit length of the string. The mass per unit length is easy to find: it's just the total mass of the string divided by the length. However, the mass of the rope isn't given -- the weight is given. So, first, we need to find the mass of the string from the weight:

w = mg --> m = w/g = (20.4)/(9.8) = 2.08 kg

Now that we know the mass, mu is

mu = m/L = (2.08)/(6) = 0.35 kg/m

Since the rope doesn't change between the different parts of the problem, this number is going to be a constant. The only thing that changes between the different problems is the tension in the rope.

At any point along the rope, the tension is going to equal the total weight that point on the rope is supporting. That total weight will always be the weight hanging at the bottom of the rope plus the weight of the rope below the point. For instance, at the bottom of the rope, the tension is just the weight of the block. Midway up the rope, it's the weight of the block plus the weight of half the rope. At the top of the rope, it's the weight of the block plus the weight of the entire rope.

Part A

At the bottom of the rope, the tension is just the weight of the block:

wB = mB*g = (0.51)(9.8) = 5 N --> T = 5N

So, the speed of transverse waves on this rope is

v = sqrt(T/mu) = sqrt((5)/(0.35)) = sqrt(14.3) = 3.78 m/s

Part B

At the middle of the rope, the tension is the weight of the block plus half the weight of the rope:

T = w_B + 1/2w = (5) + 1/2(20.4) = 15.2 N

So, the speed of the transverse waves are

v = sqrt(T/mu) = sqrt((15.2)/(0.35)) = sqrt(43.4) = 6.59 m/s

Part C

At the top of the rope, the tension is the weight of the block plus the weight of the entire rope:

T = w_B + w = (5) + (20.4) = 25.4 N

So, the speed of transverse waves is

v = sqrt(T/mu) = sqrt((25.4)/(0.35)) = sqrt(72.6) = 8.52 m/s

Part D

The average tension in the rope is the tension at the bottom of the rope plus the tension at the top, divided by two:

T_av = (25.4 + 5)/2 = 15.2 N

which is absolutely the tension at the middle of the rope (see part B). The average of the wave speeds is found the same way:

v_av = (3.78 + 8.52)/2 = 6.15 m/s

which is not equal to the speed at the middle of the rope (see part B).

answered by @doug • almost 2 years ago