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
(1) I do not understand the meaning of your
{\r and ^\circ symbols.
(2) What is your question?
(3) The subject is physics, not physic
(4) Use conservation of energy to solve the problem, if it is how high the mass goes up the slope.
degrees
To determine what happens after the spring is released, we need to analyze the forces acting on the mass placed at the free end of the spring.
1. Calculate the gravitational force acting on the mass:
The gravitational force can be calculated using the equation F_gravity = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = (2.1 kg) * (9.8 m/s^2) = 20.58 N.
2. Decompose the weight force into components:
Since the inclined plane makes an angle of 41 degrees with respect to the horizontal, we need to calculate the components of the gravitational force acting parallel and perpendicular to the slope.
F_perpendicular = F_gravity * cos(41°)
F_parallel = F_gravity * sin(41°)
3. Calculate the force exerted by the compressed spring:
The force exerted by the spring can be calculated using Hooke's Law, which states that the force is proportional to the displacement from equilibrium length: F_spring = -k * x.
Here, k represents the spring constant (80 N/m), and x represents the displacement from the equilibrium length (0.5 m - 1.0 m = -0.5 m).
F_spring = -(80 N/m) * (-0.5 m) = 40 N.
4. Analyze the forces acting on the mass:
Since the slope is frictionless, the forces acting on the mass are the spring force (F_spring), the parallel component of the weight (F_parallel), and the normal force (N) acting perpendicular to the slope.
The normal force (N) cancels out the perpendicular component of the weight (F_perpendicular), allowing us to ignore those forces in our analysis.
5. Resolve the forces along the slope:
We need to calculate the acceleration of the mass along the slope. To do this, we resolve the forces along the slope direction (parallel).
Net force along the slope = F_parallel - F_spring
Net force along the slope = 20.58 N - 40 N = -19.42 N
(Note: The negative sign indicates that the force is acting in the opposite direction of the positive direction along the slope.)
6. Calculate the acceleration:
The net force along the slope is equal to the mass multiplied by its acceleration: Net force along the slope = m * a
-19.42 N = (2.1 kg) * a
Acceleration (a) = -9.25 m/s^2
(Note: The negative sign indicates that the acceleration is opposite to the direction of the positive direction along the slope.)
Based on the calculations, the mass will experience an acceleration of -9.25 m/s^2 along the slope when the spring is released.