1. A mass of 20 kg is suspended at the end of a rubber cord of diameter 2.0 mm
and length 100 cm. Find the period of vertical oscillations of the mass.
[Take young’s modulus E, for the rubber = 2.0 x 108 Nm - 2].
To find the period of vertical oscillations of the mass, we can use the formula for the period of a simple pendulum.
The formula for the period of a simple pendulum is given by:
T = 2π√(L/g)
Where:
T = Period of oscillation (measured in seconds)
π = Pi, approximately 3.14159
L = Length of the pendulum (measured in meters)
g = Acceleration due to gravity (approximately 9.8 m/s^2)
In this case, the mass is suspended at the end of a rubber cord, which behaves like a simple pendulum. The length of the pendulum is the length of the rubber cord, which is given as 100 cm (or 1.0 meters).
However, we need to take into account the diameter of the rubber cord, which affects the effective length of the pendulum. The diameter of the cord is given as 2.0 mm.
To find the effective length, we need to subtract twice the diameter from the total length of the cord. This is because the diameter accounts for the thickness on both sides of the cord.
Effective length (L') = Length of the cord - 2 x Diameter
= 1.0 meters - 2 x (0.002 meters)
Now that we have the effective length, we can calculate the period using the formula for a simple pendulum:
T = 2π√(L/g)
Substituting the values:
T = 2π√(L'/g)
= 2π√((1.0 meters - 2 x 0.002 meters)/9.8 m/s^2)
Calculate the value inside the square root first:
(1.0 meters - 2 x 0.002 meters)/9.8 m/s^2
= 0.996/9.8 m/s^2
= 0.1018 seconds^2
Now, substitute this value back into the equation:
T = 2π√(0.1018 seconds^2)
= 2π x 0.319 seconds
= 2.0054 seconds
Therefore, the period of vertical oscillations of the mass is approximately 2.01 seconds.
To find the period of vertical oscillations of the mass, we need to use the equation for the period of a simple harmonic oscillator:
T = 2π√(m/k)
Where:
T is the period of oscillations
m is the mass of the object
k is the effective spring constant
First, let's find the value of k for the rubber cord. The formula for the effective spring constant of a cord is:
k = (π / 4) * (d^2 * E) / L
Where:
d is the diameter of the cord
E is the Young's modulus of the rubber
L is the length of the cord
Given:
d = 2.0 mm = 0.002 m
E = 2.0 x 10^8 Nm^2
L = 100 cm = 1.0 m
Now, substitute the given values into the formula for k:
k = (π / 4) * (0.002^2 * 2.0 x 10^8) / 1.0
k = (3.14 / 4) * (0.000004 * 2.0 x 10^8) / 1.0
k = (3.14 / 4) * (0.000008 x 2.0 x 10^8) / 1.0
k = (3.14 / 4) * (0.000016 x 10^8) / 1.0
k = (3.14 / 4) * (0.00016 x 10^7)
k = (3.14 / 4) * 1.6 x 10^6
k = 1.239 x 10^6 Nm^(-1)
Now, substitute the values of m and k into the equation for the period T:
T = 2π√(20 / 1.239 x 10^6)
Simplifying and evaluating the expression, we get:
T = 2π * √(20 / 1.239 x 10^6)
T = 2π * √(20) / √(1.239 x 10^6)
T = 2π * √(20) / (1112.05)
Now, approximate and calculate the value of T:
T ≈ 2π * √(20) / (1112.05)
T ≈ 6.28 * 4.472 / (1112.05)
T ≈ 28.00696 / 1112.05
T ≈ 0.0252 s
Therefore, the period of vertical oscillations of the mass is approximately 0.0252 seconds.