A 2.1 kg mass is connected to a spring with spring constant k= 180 N/m and unstretched length 16 cm. The pair are mounted on a frictionless air table, with the free end of the spring attached to a frictionless pivot. The mass is set into circular motion at 1.3 m/s. Find the radius of its path
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To find the radius of the path of the mass, we need to consider the forces acting on it.
1. Centripetal Force:
The centripetal force required to keep the mass in circular motion is provided by the tension in the spring. This force can be calculated using the equation:
Fc = mv^2 / r
Where Fc is the centripetal force, m is the mass, v is the velocity, and r is the radius.
2. Spring Force:
The spring force is given by Hooke's Law:
Fs = -kx
Where Fs is the spring force, k is the spring constant, and x is the displacement from the equilibrium position.
Since the mass is in equilibrium, the spring force balances the centripetal force:
-kx = mv^2 / r
We know the mass (m = 2.1 kg), the spring constant (k = 180 N/m), and the unstretched length of the spring (16 cm = 0.16 m).
To find the displacement x, we can use the following equation:
x = r - L
Where L is the unstretched length of the spring.
Substituting the values and solving the equation will give us the radius of the path (r).
To find the radius of the mass's circular path, we can use the concept of centripetal force and equate it to the force provided by the spring.
The centripetal force acting on an object moving in a circle of radius "r" and speed "v" is given by the formula:
Fc = mv^2 / r
where Fc is the centripetal force, m is the mass, v is the speed, and r is the radius of the circular path.
In this case, the centripetal force is provided by the spring force, which follows Hooke's Law:
Fs = -kx
where Fs is the spring force, k is the spring constant, and x is the displacement of the mass from its equilibrium position.
Since the mass is attached to the spring and moving in a circle, the displacement x is equal to the stretched length of the spring, which is given by:
x = R - L
where R is the radius of the circular path and L is the unstretched length of the spring.
We can combine the equations by equating the centripetal force to the spring force:
mv^2 / r = -k(R - L)
Rearranging the equation, we get:
R = (mv^2 + kL) / k
Given that the mass (m) is 2.1 kg, the speed (v) is 1.3 m/s, the spring constant (k) is 180 N/m, and the unstretched length (L) is 16 cm, or 0.16 m, we can substitute these values into the equation to find the radius (R).
R = (2.1 kg * (1.3 m/s)^2 + 180 N/m * 0.16 m) / 180 N/m
Calculating the expression, we find:
R ≈ 0.5125 m
Therefore, the radius of the mass's circular path is approximately 0.5125 meters.