A spring S of force constant k=100 N/m. is fixed to the base of a 30 degree incline.A mass m=50g. is held against the free end of the spring,so that the spring is compressed by 0.1 metre.If the mass is now released,calculate the distance travelled by the mass up the incline.ignore friction
To calculate the distance traveled by the mass up the incline, we need to analyze the potential energy stored in the compressed spring and compare it to the potential energy gained by the mass as it moves up the incline.
1. Calculate the potential energy stored in the compressed spring:
The potential energy stored in a spring is given by the equation U = (1/2)kx^2, where U is the potential energy, k is the force constant, and x is the displacement from the equilibrium position.
In this case, the force constant of the spring is k = 100 N/m and the displacement is x = 0.1 m.
Substituting these values into the equation, we have U = (1/2)(100 N/m)(0.1 m)^2 = 0.5 J.
2. Calculate the potential energy gained by the mass:
The potential energy gained by the mass as it moves up the incline is given by the equation U = mgh, where U is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height gained.
In this case, the mass of the object is m = 50 g = 0.05 kg.
The angle of the incline is 30 degrees, so the height gained can be calculated as h = x*sin(theta), where x is the displacement along the incline.
The hypotenuse can be calculated as hypotenuse = x/cos(theta).
Then, the height gained is h = hypotenuse*sin(theta) - hypotenuse.
Substituting the values into the equation, we have U = (0.05 kg)(9.8 m/s^2)(x*sin(theta) - x/cos(theta)).
3. Equate the two potential energies:
Equating the potential energies from the spring and the mass, we have 0.5 J = (0.05 kg)(9.8 m/s^2)(x*sin(theta) - x/cos(theta)).
4. Solve for x:
Rearranging the equation to solve for x, we have x = 0.5 J / [(0.05 kg)(9.8 m/s^2)(sin(theta) - 1/cos(theta))].
Substituting the value of theta (30 degrees) into the equation, we can calculate the distance traveled by the mass up the incline.