A spring (80 {\rm N/m}) has an equilibrium length of 1.00 {\rm m}. The spring is compressed to a length of 0.50 {\rm m} and a mass of 2.1 {\rm kg} is placed at its free end on a frictionless slope which makes an angle of 41 ^\circ with respect to the horizontal. The spring is then released
To determine what happens when the spring is released, we need to analyze the forces acting on the mass and determine the resulting motion.
1. Calculate the gravitational force acting on the mass:
The gravitational force is given by the equation F_gravity = m * g, where m is the mass (2.1 kg) and g is the acceleration due to gravity (9.8 m/s²). Plugging the values, we get:
F_gravity = 2.1 kg * 9.8 m/s² = 20.58 N
2. Resolve the gravitational force into components:
Since the slope makes an angle of 41° with respect to the horizontal, we need to determine the component of the gravitational force acting parallel to the slope (F_parallel) and the component acting perpendicular to the slope (F_perpendicular).
F_parallel = F_gravity * sin(41°)
F_parallel = 20.58 N * sin(41°) = 13.44 N
F_perpendicular = F_gravity * cos(41°)
F_perpendicular = 20.58 N * cos(41°) = 15.67 N
3. Determine the net force acting on the mass:
The net force can be calculated by considering the force due to the compressed spring (F_spring) and the component of the gravitational force parallel to the slope (F_parallel).
F_net = F_spring + F_parallel
4. Calculate the force exerted by the compressed spring (F_spring):
The force exerted by a spring is given by Hooke's Law, F_spring = -k * x, where k is the spring constant (80 N/m) and x is the extension or compression from the equilibrium position. Since the spring is compressed by 0.5 m from its equilibrium length of 1.0 m, we have:
F_spring = -80 N/m * (-0.5 m) = 40 N
5. Determine the net force:
F_net = F_spring + F_parallel
F_net = 40 N + 13.44 N = 53.44 N
6. Determine the acceleration of the mass:
The net force on the mass causes acceleration. Using Newton's second law, F_net = m * a, we can calculate the acceleration (a):
53.44 N = 2.1 kg * a
a = 53.44 N / 2.1 kg ≈ 25.45 m/s²
Since the acceleration is positive, the mass will accelerate down the slope when the spring is released.
Note: This analysis assumes ideal conditions without any air resistance or other factors that could affect the motion.