5. A 2.00-kg block hangs from a rubber cord, being supported so that the cord is not stretched. The unstretched length of the cord is 0.500 m, and its mass is 5.00 g. The “spring constant” for the cord is 100 N/m. The block is released and stops momentarily at the lowest point.(a) Determine the tension in the cord when the block is at this lowest point. (b) What is the length of the cord in this “stretched” position? (c) If the block is held in this lowest position, find the speed of a transverse wave in the cord.
dede
To solve this problem, we need to use the principles of simple harmonic motion and the equation for the tension in a spring.
(a) To determine the tension in the cord when the block is at the lowest point, we first need to calculate the extension or compression of the cord. Since the block stops momentarily at the lowest point, the extension of the cord is equal to the initial length of the cord minus the stretched length when the block is at rest.
Given:
Unstretched length of the cord (l0) = 0.500 m
Mass of the cord (m) = 5.00 g = 0.005 kg
Spring constant (k) = 100 N/m
The extension or compression of the cord (Δl) can be calculated using Hooke's Law:
F = k * Δl
Where:
F is the force (tension) in the cord
k is the spring constant
Δl is the extension or compression of the cord
Rearranging the equation, we have:
Δl = F / k
Substituting the known values, we get:
Δl = (m * g) / k
Since the block is at rest, the force acting on it is its weight, given by:
Weight = m * g
Substituting the values, we have:
Δl = Weight / k = (0.005 kg * 9.8 m/s²) / 100 N/m
Solving this equation will give us the extension or compression (Δl) of the cord.
(b) The length of the cord in the stretched position is equal to the sum of the initial unstretched length (l0) and the extension or compression (Δl).
Stretched length = l0 + Δl
(c) To find the speed of a transverse wave in the cord, we first need to calculate the mass per unit length (μ) of the cord. This is given by:
μ = m / l0
Where:
μ is the mass per unit length of the cord
m is the mass of the cord
l0 is the unstretched length of the cord
Once we have the mass per unit length, the speed of a transverse wave (v) in the cord is given by:
v = √(T/μ)
Where:
v is the speed of the wave
T is the tension in the cord
μ is the mass per unit length
Substituting the known values, we can calculate the speed of the transverse wave.
Let's go through the calculations step by step:
To find the answers to these questions, we need to use some basic principles of physics related to springs and waves. Let's go step by step:
(a) To determine the tension in the cord when the block is at the lowest point, we need to consider the forces acting on the block. At the lowest point, the tension in the cord should balance the weight of the block.
The weight of the block can be calculated using the equation:
Weight = mass * gravitational acceleration
Weight = 2.00 kg * 9.8 m/s^2
Weight = 19.6 N
Since the block is at rest at the lowest point, the tension in the cord should be equal to the weight. Therefore, the tension in the cord is 19.6 N.
(b) To find the length of the cord in this "stretched" position, we need to consider the extension of the cord when the block is hanging. The extension of the cord is calculated by the equation:
Extension = (force applied / spring constant)
In this case, the force applied is the weight of the block, which is 19.6 N. And the spring constant is given as 100 N/m.
Extension = 19.6 N / 100 N/m
Extension = 0.196 m
To find the length of the cord in the stretched position, we add the extension to the original unstretched length of the cord:
Length of cord in stretched position = unstretched length + extension
Length of cord in stretched position = 0.500 m + 0.196 m
Length of cord in stretched position = 0.696 m
Therefore, the length of the cord in the stretched position is 0.696 m.
(c) To find the speed of a transverse wave in the cord, we need to use the equation:
Wave speed = sqrt(tension / linear mass density)
The tension in the cord is 19.6 N, which we found in part (a). The linear mass density is the mass of the cord divided by its length.
The mass of the cord is given as 5.00 g, which can be converted to kg by dividing by 1000 (1 g = 0.001 kg). The length of the cord is 0.696 m, which we found in part (b).
Linear mass density = mass / length
Linear mass density = 5.00 g / 0.696 m
Linear mass density = 7.184 kg/m
Plugging these values into the equation for wave speed:
Wave speed = sqrt(19.6 N / 7.184 kg/m)
Wave speed ≈ 5.157 m/s
Therefore, the speed of the transverse wave in the cord is approximately 5.157 m/s.