A mass, m1 = 9.36 kg, is in equilibrium while connected to a light spring of constant k = 112 N/m that is fastened to a wall.
A second mass, m2 = 6.71 kg, is slowly pushed up against mass m1, compressing the spring by the amount A = 0.201 m
The system is then released, and both masses start moving to the right on the frictionless surface. When m1 reaches the equilibrium point, m2 loses contact with m1 and moves to the right with speed v. Determine the value of v.
At the spring's equilbrium point, all of the stored potential energy of the compressed spring becomes kinetic energy of both masses. No further force is applied by the spring at that instant, and then the masses separate as the spring force reverses direction.
The V (of both masses) at that instant is given by
(1/2) k A^2 = (1/2)(m1 + m2) V^2
Solve for V
To determine the value of v, we need to use the principle of conservation of energy.
1. First, we need to find the potential energy stored in the compressed spring. The equation for potential energy stored in a spring is given by: PE = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position. Here, k = 112 N/m and x = 0.201 m.
Substituting these values into the equation, we have:
PE = (1/2)(112 N/m)(0.201 m)^2
2. Next, we need to calculate the potential energy converted to kinetic energy when mass m1 reaches the equilibrium point. At the equilibrium point, all the potential energy stored in the spring is converted to kinetic energy. The equation for kinetic energy is given by: KE = (1/2)mv^2, where m is the mass and v is the velocity.
Since kinetic energy is conserved, it can be equated to the potential energy stored in the spring:
(1/2)mv^2 = (1/2)(112 N/m)(0.201 m)^2
3. Now, we can solve the equation for v by rearranging and plugging in the known values:
v^2 = (112 N/m)(0.201 m)^2 / m
v^2 = 4.512 Nm^2/kg
4. Finally, taking the square root of both sides gives us the value of v:
v = √(4.512 Nm^2/kg)
Calculating the square root of 4.512 Nm^2/kg, we find:
v = 2.122 m/s
Therefore, the value of v is 2.122 m/s.