A 15 kg mass, tied to the end of a 0.5 m long steel wire, is whirled in a vertical circle. The cross- section of the wire is 0.02 cm2. When the mass is at the bottom of the circle, its angular velocity is 2 rev s i. Find the elongation of the wire at this point.
Y=2•10¹¹ Pa
ω=2πf =2π•2=4π rad/s.
ma=T-mg
m ω²R =T – mg
T=m(ω²R+g) =m(16π²R +g)=…
T/A =Y•ΔL/L
ΔL=TL/AY =…
To find the elongation of the wire at the bottom of the vertical circle, we can use the concept of centripetal force.
First, let's calculate the tension in the wire at the bottom of the circle using the centripetal force formula:
Centripetal force (Fc) = Mass (m) x Acceleration (a)
Since the motion is in a vertical circle, the centripetal force is provided by the tension in the wire and the force due to gravity:
Fc = Tension (T) - Weight (mg)
where m is the mass and g is the acceleration due to gravity.
Now, let's find the values for these variables:
Mass (m) = 15 kg (given in the problem)
Acceleration due to gravity (g) = 9.8 m/s^2 (standard value)
Now, let's solve for the tension (T) at the bottom of the circle:
Fc = T - mg
T = Fc + mg
Centripetal force (Fc) can be derived from the angular velocity (ω) using the formula:
Fc = m * ω^2 * r
ω is the angular velocity, which is given as 2 revolutions per second. To convert revolutions to radians, we multiply by 2π:
ω = 2 rev/s * 2π rad/rev = 4π rad/s
The length of the wire (r) is given as 0.5 m.
Now we can substitute the values and calculate the tension (T):
Fc = m * ω^2 * r
T = Fc + mg
T = (m * ω^2 * r) + (m * g)
Plugging in the values:
T = (15 kg * (4π rad/s)^2 * 0.5 m) + (15 kg * 9.8 m/s^2)
T ≈ 47.1 N
Now that we have the tension (T), we can calculate the elongation of the wire at the bottom of the circle.
The elongation can be found using Hooke's Law:
Tension (T) = k * x
where T is the tension, k is the spring constant, and x is the elongation.
Since the cross-section of the wire (A) is given as 0.02 cm^2, we can find the spring constant (k) using Young's modulus (Y) formula:
k = Y * A / L
where Y is the Young's modulus and L is the original length of the wire.
Young's modulus (Y) for steel is approximately 2.0 x 10^11 N/m^2 (standard value).
Now let's substitute the values into the formula for k:
k = (2.0 x 10^11 N/m^2) * (0.02 x 10^-4 m^2) / 0.5 m
k ≈ 8.0 x 10^6 N/m
Finally, we can calculate the elongation (x):
Tension (T) = k * x
x = T / k
Plugging in the values:
x = (47.1 N) / (8.0 x 10^6 N/m)
x ≈ 5.9 x 10^-6 m
Therefore, the elongation of the wire at the bottom of the circle is approximately 5.9 x 10^-6 meters.