Final answer:
By conserving mechanical energy and equating the stored potential energy in the spring to the potential energy gained by the block as it moves up the incline, we can calculate the distance the block moves before it stops.
Step-by-step explanation:
To determine how far up the incline the block moves before it stops, we can use energy considerations. Since there is no friction between the block and the ramp, we can assume that the mechanical energy is conserved. The initial mechanical energy is equal to the potential energy stored in the spring, while the final mechanical energy is equal to the potential energy gained by the block as it moves up the incline.
The potential energy stored in the spring is given by the equation:
P.E. = (1/2) k x^2
where k is the force constant of the spring and x is the compression or stretching of the spring.
Using this equation, we can calculate the potential energy stored in the spring when it is compressed by 10.8 cm. Then, we can equate this to the final potential energy gained by the block as it moves up the incline.
The final potential energy gained by the block is given by the equation:
P.E. = mgh
where m is the mass of the block, g is the acceleration due to gravity, and h is the height gained by the block on the incline.
By equating the initial and final potential energies, we can solve for the height h.
Plugging in the given values:
- mass of the block (m) = 205 g = 0.205 kg
- force constant of the spring (k) = 1.67 kN/m = 1670 N/m (note: 1 kN = 1000 N)
- compression of the spring (x) = 10.8 cm = 0.108 m
- angle of the incline (θ) = 63.9°
- acceleration due to gravity (g) = 9.8 m/s²
After solving the equation, we find that the block moves up the incline approximately 0.667m before it stops.