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 solve this problem, we need to calculate the potential energy of the compressed spring, find the height of the slope, and determine the speed of the mass as it reaches the bottom of the slope.
Step 1: Calculate the potential energy of the compressed spring.
The potential energy stored in the spring is given by the formula:
PE = (1/2)kx^2
where PE is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
Given:
spring constant, k = 70 N/m
displacement, x = 1.00 m - 0.50 m = 0.50 m
Plugging in the values:
PE = (1/2)(70 N/m)(0.50 m)^2
PE = (1/2)(70 N/m)(0.25 m^2)
PE = 8.75 J
Step 2: Find the height of the slope.
The potential energy stored in the spring is converted into gravitational potential energy as the mass moves up the slope. The gravitational potential energy is given by the formula:
PE = mgh
where PE is the potential energy, m is the mass, g is the acceleration due to gravity, and h is the height of the slope.
Given:
mass, m = 2.1 kg
slope angle, θ = 41°
acceleration due to gravity, g = 9.8 m/s^2
To find h, we need to find the vertical component of the slope's length using trigonometry:
h = L * sin(θ)
where L is the length of the slope.
Using the given displacement x = 0.50 m, we can find L:
L = x / cos(θ)
Plugging in the values:
L = 0.50 m / cos(41°)
L ≈ 0.635 m
Now we can calculate h:
h = 0.635 m * sin(41°)
h ≈ 0.408 m
Step 3: Determine the speed of the mass at the bottom of the slope.
Using the conservation of mechanical energy, we can equate the initial potential energy stored in the spring to the final kinetic energy of the moving mass.
Initial potential energy = Final kinetic energy
The initial potential energy is given by the formula:
PE = mgh
The final kinetic energy is given by the formula:
KE = (1/2)mv^2
where KE is the kinetic energy, m is the mass, and v is the velocity/speed.
Given:
mass, m = 2.1 kg
height, h = 0.408 m
acceleration due to gravity, g = 9.8 m/s^2
Plugging in the values for the initial potential energy:
PE = mgh = (2.1 kg)(9.8 m/s^2)(0.408 m)
PE ≈ 8.758 J
Now we can solve for the final kinetic energy and then find the speed:
8.758 J = (1/2)(2.1 kg)v^2
Solving for v:
v^2 = (2 * 8.758 J) / (2.1 kg)
v^2 ≈ 8.324
v ≈ √8.324 ≈ 2.889 m/s
Therefore, the speed of the mass at the bottom of the slope is approximately 2.889 m/s.
To determine the motion of the spring when it is released, we need to consider the forces acting on the system.
1. Gravitational force: The mass placed at the free end of the spring experiences a gravitational force acting downward. The magnitude of this 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).
2. Normal force: The mass also experiences a normal force perpendicular to the slope. The magnitude of the normal force can be calculated using the equation: F_normal = m * g * cos(theta), where theta is the angle of the slope with respect to the horizontal.
3. Spring force: The compressed spring exerts a force in the opposite direction of compression. The magnitude of this 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 length (1.00 m - 0.50 m = 0.50 m).
4. Friction force: Since the slope is frictionless, there is no friction force acting on the mass.
Now, we can analyze the forces along the slope and perpendicular to the slope:
Along the slope:
- The gravitational force component parallel to the slope is F_gravity_parallel = m * g * sin(theta).
- The net force along the slope is the difference between the component of the gravitational force and the component of the spring force acting along the slope. The net force along the slope is F_net_parallel = F_gravity_parallel - F_spring.
Perpendicular to the slope:
- The normal force is equal to the gravitational force component perpendicular to the slope: F_normal = m * g * cos(theta).
Using these forces, we can determine the acceleration of the mass along the slope using Newton's second law: F_net_parallel = m * a, where a is the acceleration.
To sum up, the key steps to analyze the motion of the spring released on a frictionless slope are:
1. Calculate the gravitational force and the normal force.
2. Determine the component of the gravitational force and the spring force acting along the slope.
3. Calculate the net force along the slope.
4. Determine the acceleration using Newton's second law.
5. Analyze the motion of the mass using the calculated acceleration.
Note: The specific details of the motion, such as the direction and distance traveled, can be further analyzed depending on the specific context and constraints of the problem.