The string passes over a frictionless pulley

Consider what is shown below. A 93-kg mass M rests on a frictionless ramp inclined at 11°. 

A string with a linear mass density of μ=0.0012 kg/m is attached to the mass M. 

The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass m. 

The system is in static equilibrium. 

A wave is induced on the string and travels up the ramp. 

At what wave speed in m/s does the wave travel up the string? 

Input the numerical part of your answer in electronic E-scientific notation using 3 significant figures. Use: g = 9.8 m/s2

To solve the problem of determining the wave speed at which a wave travels up the string in the described scenario, we need to use the equation for the wave speed on a string, which is given by: 

v=μT​​

where

v is the wave speed,
T is the tension in the string, and
μ is the linear mass density of the string.
Calculate the tension in the string (T):
The tension is due to the weight of the hanging mass m.
Since the system is in static equilibrium,
the component of the gravitational force along the incline
that the mass M on the ramp must balance is equal to the tension T in the string.
The force along the incline due to M is M g sin(θ),
where
g is the acceleration due to gravity and
θ is the angle of the incline.
Thus, we have:

$$�=��\sin (�)$$

2. Linear mass density of the string ($�$): Given directly as $0.0012\text{kg/m}$.

3. Substitute and calculate $�$: Substitute the known values into the wave speed formula to calculate $�$.

Given values:

• $�=93\text{kg}$
• $�=11^{\circ }$
• $�=0.0012\text{kg/m}$
• $�=9.8{\text{m/s}}^2$

Let's perform these calculations.

The wave speed at which the wave travels up the string is approximately $380.683\text{m/s}$.

Expressed in electronic E-scientific notation with three significant figures, this is $3.81\times 10^2\text{m/s}$.