A spring (70 {\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 analyze the motion of the spring, we need to consider the forces acting on the mass.
1. Gravitational force: The mass (2.1 kg) experiences a gravitational force pulling it downwards. The magnitude of this force can be calculated using the equation: F_grav = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2). So, F_grav = 2.1 kg * 9.8 m/s^2 = 20.58 N.
2. Normal force: The component of the gravitational force perpendicular to the slope is balanced by the normal force acting on the mass. The normal force has the same magnitude as the component of the gravitational force perpendicular to the slope, which can be calculated as F_perp = F_grav * cos(angle), where the angle is 41 degrees. So, F_perp = 20.58 N * cos(41) = 15.52 N.
3. Spring force: As the spring is compressed from its equilibrium position, it exerts a force to restore itself to its equilibrium length. The magnitude of the spring force can be calculated using Hooke's Law: F_spring = -k * x, where k is the spring constant (70 N/m) and x is the displacement from the equilibrium position (0.50 m - 1.00 m = -0.50 m). So, F_spring = -70 N/m * (-0.50 m) = 35 N.
Now, let's analyze the forces along the slope direction:
1. Parallel force: The component of the gravitational force parallel to the slope will cause the mass to accelerate down the slope. The magnitude of this force can be calculated as F_parallel = F_grav * sin(angle), where the angle is 41 degrees. So, F_parallel = 20.58 N * sin(41) = 13.07 N.
2. Frictional force: Assuming the slope is frictionless, there is no frictional force acting on the mass in the parallel direction.
Since the only force acting in the parallel direction is the component of the gravitational force, the mass will accelerate down the slope. The acceleration can be calculated using Newton's second law: F_parallel = m * a, where m is the mass (2.1 kg) and a is the acceleration. So, 13.07 N = 2.1 kg * a. Solving for a, we get a = 13.07 N / 2.1 kg = 6.22 m/s^2.
Therefore, the mass will accelerate down the slope with an acceleration of 6.22 m/s^2.