A spring with a force constant of 1.0 N/cm is compressed to a displacement of 10.0 cm. When released, it's energy is used to rotate a mass of 0.50 kg, secured by a spring, around in a circle with radius of 0.75 m, starting at the top of the circle. After 0.3 s has elapsed, the string is cut. The bottom of the circle is 0.5 m above the ground.
To solve this problem, we need to consider the conservation of energy and use the formulas related to spring potential energy and circular motion.
First, let's find the potential energy stored in the compressed spring.
The formula for potential energy stored in a spring is given by:
Potential Energy (PE) = (1/2) * k * x^2
Where:
k is the force constant of the spring (1.0 N/cm = 100 N/m because 1 N/cm = 100 N/m)
x is the displacement of the spring (10.0 cm = 0.1 m)
Substituting the values, we have:
PE = (1/2) * 100 N/m * (0.1 m)^2
PE = 0.5 N * 0.01 m
PE = 0.005 J
So, the potential energy stored in the compressed spring is 0.005 Joules.
Next, let's calculate the kinetic energy of the rotating mass after it is released. At the top of the circle, the mass has potential energy which converts into kinetic energy as it moves down.
The potential energy at the top of the circle, just before being released, is given by:
Potential Energy (PE) = mass * gravity * height
Where:
mass (m) = 0.50 kg
gravity (g) = 9.8 m/s^2
height = 0.5 m
Substituting the values, we have:
PE = 0.50 kg * 9.8 m/s^2 * 0.5 m
PE = 2.45 J
Since energy is conserved, the potential energy at the top will convert to kinetic energy at the bottom of the circle:
KE = PE = 2.45 J
Now, let's calculate the speed of the rotating mass at the bottom of the circle.
Kinetic Energy (KE) = (1/2) * mass * velocity^2
Substituting the values, we have:
2.45 J = (1/2) * 0.50 kg * velocity^2
velocity^2 = 2.45 J / (0.50 kg * 0.5)
velocity^2 = 9.8 m^2/s^2
Taking the square root of both sides, we get:
velocity = sqrt(9.8 m^2/s^2)
velocity = 3.13 m/s (approximately)
So, the speed of the rotating mass at the bottom of the circle is approximately 3.13 m/s.
Finally, let's determine the centripetal force acting on the rotating mass.
The centripetal force required to keep an object moving in a circle is given by:
Centripetal Force (F) = (mass * velocity^2) / radius
Where:
mass (m) = 0.50 kg
velocity = 3.13 m/s
radius = 0.75 m
Substituting the values, we have:
F = (0.50 kg * (3.13 m/s)^2) / 0.75 m
F = (0.50 kg * 9.80 m^2/s^2) / 0.75 m
F = 6.53 N (approximately)
So, the centripetal force acting on the rotating mass is approximately 6.53 Newtons.