Spring-Mass System Dynamics

The video tutorial explains the setup of a mechanical system with two masses and springs, detailing the forces acting on each mass. It covers the derivation of equations of motion, solving them using matrices, and finding eigenvalues and eigenvectors. The tutorial concludes with determining the general solution and applying initial conditions to find specific solutions, illustrating the natural modes of oscillation.

What are the masses of the two objects in the spring-mass system?

• 3 kg and 1 kg

• 2 kg and 1 kg

• 2 kg and 3 kg

• 1 kg and 2 kg

Which equation represents the force exerted by a spring?

• Force = mass × velocity

• Force = spring constant × displacement

• Force = mass × displacement

• Force = spring constant × velocity

What is the spring constant K1 in the system?

• 2 N/m

• 4 N/m

• 8 N/m

• 6 N/m

How is the force on mass 1 from spring 1 described?

• Negative K1 times X1

• Positive K2 times X2

• Positive K1 times X1

• Negative K2 times X2

What is the form of the matrix equation for the system?

• M × X = K × X double prime

• M × X double prime = K × X

• K × X = M × X double prime

• X double prime = M × K × X

What are the eigenvalues of the matrix A?

• -2 and -3

• -1 and -4

• 1 and 4

• 2 and 3

What is the general solution of the system based on eigenvalues?

• X(t) = V1 × (A1 sin(2t) + B1 cos(2t)) + V2 × (A2 sin(t) + B2 cos(t))

• X(t) = V1 × (A1 cos(2t) + B1 sin(2t)) + V2 × (A2 cos(t) + B2 sin(t))

• X(t) = V1 × (A1 sin(t) + B1 cos(t)) + V2 × (A2 sin(2t) + B2 cos(2t))

• X(t) = V1 × (A1 cos(t) + B1 sin(t)) + V2 × (A2 cos(2t) + B2 sin(2t))

What are the natural frequencies of the system?

• 1 and 3

• 2 and 4

• 1 and 2

• 3 and 4

What are the initial conditions used to find the particular solution?

• X(0) = [0, 1], X'(0) = [6, 0]

• X(0) = [1, -1], X'(0) = [0, 6]

• X(0) = [1, 1], X'(0) = [6, 6]

• X(0) = [-1, 1], X'(0) = [0, -6]

How are the two modes of oscillation represented in the solution?

• As the sum of sine and cosine functions with different frequencies

• As the sum of two cosine functions

• As the sum of two sine functions

• As the product of sine and cosine functions