At the top of an inclined track, a spring is attached to a rod at one end and a dynamics cart at the other such that the spring stretches parallel to the track. The cart has a mass of .850kg and there is negligible friction between the cart and the track. The spring has a force constant of 12.5 N/m. Determine the stretch length of the string if the track makes an angle of 12 degrees above the horizontal.
To determine the stretch length of the spring, first, we need to calculate the gravitational force and the component of the gravitational force parallel to the track. Then, we can equate this force to the force exerted by the spring to find the stretch length.
1. Calculate the gravitational force:
The gravitational force acting on the cart can be calculated using the mass (m) and the acceleration due to gravity (g). The formula is:
F_gravity = m * g
Given m = 0.850 kg (mass of the cart) and g = 9.8 m/s^2 (acceleration due to gravity), we have:
F_gravity = 0.850 kg * 9.8 m/s^2 = 8.33 N
2. Calculate the component of the gravitational force parallel to the track:
To find the component of the gravitational force acting parallel to the track, we need to calculate the sine of the angle between the horizontal and the track. The formula is:
F_parallel = F_gravity * sin(angle)
Given angle = 12 degrees, we have:
F_parallel = 8.33 N * sin(12 degrees) = 1.78 N
3. Equate the force exerted by the spring to the parallel component of the gravitational force:
According to Hooke's Law, the force exerted by the spring is proportional to the stretch length (x). The formula is:
F_spring = k * x
Given the force constant (k) = 12.5 N/m, we have:
1.78 N = 12.5 N/m * x
4. Solve for the stretch length (x):
Rearranging the equation:
x = 1.78 N / 12.5 N/m = 0.1424 m
Therefore, the stretch length of the spring is approximately 0.1424 meters.