A mass m = 12.0 kg rests on a frictionless table and accelerated by a spring with spring constant k = 4597.0 N/m. The floor is frictionless except for a rough patch. For this rough path, the coefficient of friction is μk = 0.49. The mass leaves the spring at a speed v = 3.8 m/s.
To find the acceleration of the mass as it leaves the spring at a speed of 3.8 m/s, we can use the principle of conservation of mechanical energy.
The mechanical energy of the system remains constant, which includes the potential energy stored in the spring (Elastic Potential Energy) and the kinetic energy of the mass (Kinetic Energy).
The potential energy stored in the spring is given by the formula:
Potential Energy (PE) = (1/2) k x^2
Where k is the spring constant and x is the displacement from the equilibrium position of the spring.
The kinetic energy of the mass is given by the formula:
Kinetic Energy (KE) = (1/2) m v^2
Where m is the mass and v is the velocity of the mass.
Since there is no friction on the table, the mechanical energy of the system before and after the mass leaves the spring should be the same. Therefore, we can equate the potential energy of the spring to the kinetic energy of the mass:
(1/2) k x^2 = (1/2) m v^2
Now we need to rearrange the equation to solve for x, the displacement from the equilibrium position.
x^2 = (m v^2) / k
x = sqrt( (m v^2) / k)
By substituting the given values into the equation, we can find the displacement x:
x = sqrt((12.0 kg * (3.8 m/s)^2) / 4597.0 N/m)
x = sqrt(176.16 kg·m^2/s^2 / 4597.0 N/m)
x ≈ 0.086 m
Now, to find the acceleration, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
Net Force = m * a
Since the force acting on the mass when it leaves the spring is the force exerted by the spring, given by Hooke's Law:
Force = k * x
We can now substitute the values of k and x we found:
k * x = m * a
(4597.0 N/m) * 0.086 m = (12.0 kg) * a
a ≈ 391.7 m/s^2
So, the acceleration of the mass as it leaves the spring at a speed of 3.8 m/s is approximately 391.7 m/s^2.