A m = 2.51 kg mass starts from rest and slides a distance d down a frictionless theta = 34.6° incline. While sliding, it comes into contact with an unstressed spring of negligible mass.
The mass slides an additional 0.220 m as it is brought momentarily to rest by compression of the spring (k=440 N/m). Calculate the initial separation d between the mass and the spring.
To find the initial separation d between the mass and the spring, we can use the conservation of mechanical energy principle.
1. First, let's find the potential energy at the starting position (initial height) of the mass. The potential energy (PE) can be calculated using the formula:
PE = m * g * h
where m is the mass (2.51 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the initial height.
Since the mass starts from rest, the initial kinetic energy (KE) is zero. Therefore, the total mechanical energy (E) at the starting position is:
E = PE + KE = m * g * h + 0 = m * g * h
2. Next, let's find the potential energy at the ending position (where the mass is momentarily at rest due to spring compression). The potential energy at this position can be calculated by taking into account the additional distance of 0.220 m that the mass slides down the incline, considering the gravitational potential energy and the potential energy stored in the compressed spring.
The gravitational potential energy (PE_gravity) at this position is given by:
PE_gravity = m * g * (h + d + 0.220)
where d is the initial separation we want to find.
The potential energy stored in the compressed spring (PE_spring) is given by:
PE_spring = (1/2) * k * x²
where k is the spring constant (440 N/m) and x is the distance the spring is compressed. Since the mass comes to rest momentarily, the spring potential energy equals zero:
PE_spring = 0
Therefore, the total potential energy (PE) at the ending position is:
PE = PE_gravity + PE_spring = m * g * (h + d + 0.220) + 0
3. According to the conservation of mechanical energy principle, the total mechanical energy at the starting position should be equal to the total potential energy at the ending position:
E = PE = m * g * h
Equating the expressions for E in both cases:
m * g * h = m * g * (h + d + 0.220)
Expanding the equation and solving for d:
h = (h + d + 0.220)
d = h - 0.220
Therefore, the initial separation d between the mass and the spring is equal to the initial height h minus 0.220 m.