A mass m = 13 kg is connected to two different springs, one on the left and another on the right. It is displaced a distance x to the right from its equilibrium position. (Take k1 = 115 N/m and k2 = 198 N/m.)

(a) What is the net force acting on the mass? (Use any variable or symbol stated above along with the following as necessary: k_eff for the effective spring constant.)
What is the effective spring constant?

(b) What will the frequency of the oscillation be when the mass is released?

(c) If x = 0.1 m, what is the total energy of the mass-spring system after the mass is released, and what is the maximum velocity of the mass?

Im not sure how to solve these, or where to begin. Any help or advice is appreciated, thank you!

(a)

F1= - k1•x
F2 =- k2•x
F=F1+F2= - (k1+k2) •x
k(eff) = k1+k2 =115+198=313 N/m.

(b)
ω =sqrt{k(eff)/m}=sqrt(313/13) = 4.9 rad/s.
f = ω/2•π =4.9/2•π = 0.78 Hz.
(c)
x=A=0.1 m.
E=kA²/2 =313•0.1²/2 =1.565 J.
v(max) = A• ω =0.1•4.9=0.49 m/s.

Thank you so much for your help!

To solve this problem, we will use the concepts of spring forces, Hooke's Law, and simple harmonic motion.

(a) The net force acting on the mass can be found by summing the forces from both springs. Each spring exerts a force given by Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position.

The force from the first spring (k1) is given by: F1 = -k1 * x, where x is the displacement to the right.

The force from the second spring (k2) is given by: F2 = -k2 * x, where x is the same displacement.

The net force is the sum of these forces: F_net = F1 + F2.

Now, let's calculate the net force and the effective spring constant:

F_net = (-k1 * x) + (-k2 * x)
= -(k1 + k2) * x

Therefore, the net force acting on the mass is -k_eff * x, where k_eff is the effective spring constant defined as the sum of k1 and k2: k_eff = k1 + k2.

(b) The frequency of the oscillation can be calculated using the equation: f = (1 / 2π) * √(k_eff / m), where f is the frequency, k_eff is the effective spring constant, and m is the mass.

Substituting the given values:
f = (1 / 2π) * √[(k1 + k2) / m]
= (1 / 2π) * √[(115 N/m + 198 N/m) / 13 kg]

(c) To calculate the total energy and maximum velocity of the mass, we need to know the amplitude of the oscillation. Since the amplitude is not given, we'll assume it to be equal to the displacement x.

The total energy (E) of the mass-spring system is the sum of the kinetic energy (K) and the potential energy (U).

K = (1/2) * m * v^2, where v is the velocity of the mass.

U = (1/2) * k_eff * x^2

E = K + U

To find the maximum velocity (v_max), we need to calculate the velocity when the displacement is maximum (i.e., at x = 0.1 m).

Now, let's substitute the given values and calculate:

K = (1/2) * 13 kg * v^2
U = (1/2) * (k1 + k2) * (0.1 m)^2

E = K + U

We have the equations for E and v_max, but we need to solve for v_max by substituting the value of E.

I hope this helps! Let me know if you have any further questions.

To solve these problems, we need to apply the concepts of Hooke's law and the principles of simple harmonic motion.

(a) The net force acting on the mass can be determined by considering the forces acting on it. When the mass is displaced to the right, the left spring is compressed and the right spring is extended, resulting in opposite forces. The force exerted by the left spring is given by Hooke's law: F1 = -k1x, where x is the displacement. Similarly, the force exerted by the right spring is F2 = k2x. Since the forces exerted by the springs are in opposite directions, the net force is the sum of these individual forces: F_net = F1 + F2. Substituting the given values, we have F_net = -k1x + k2x.

To find the effective spring constant, we need to consider the combination of the two springs. The effective spring constant, k_eff, is given by: 1/k_eff = 1/k1 + 1/k2. We can substitute the values of k1 and k2 to calculate k_eff.

(b) The frequency of oscillation, f, is related to the effective spring constant and the mass, m, by the equation: f = (1/2π) * √(k_eff/m). Substitute the given values of k_eff and m to calculate the frequency.

(c) To find the total energy of the mass-spring system after the mass is released, we can use the formula for the total energy in simple harmonic motion. The total energy is given by the sum of the potential energy, U, and the kinetic energy, K. The potential energy, U, is equal to (1/2) * k_eff * x^2, while the kinetic energy, K, is given by (1/2) * m * v^2, where v is the velocity of the mass. The total energy is E = U + K. Substitute the given values of x, k_eff, and m to calculate the total energy.

Since the given problem does not provide information about the initial conditions (e.g., if the mass is released from rest or with an initial velocity), we cannot directly determine the maximum velocity of the mass. However, when the mass reaches its maximum displacement, the velocity will momentarily be zero before it starts moving in the opposite direction. Therefore, the maximum velocity can be calculated as the velocity at maximum displacement, which can be determined using the appropriate equations of motion.

I hope this explanation helps you understand how to approach and solve these problems! Remember to substitute the given values into the equations to calculate the desired quantities.