The two blocks A and B each have a mass of 5 kg and are suspended from parallel cords. A spring,

The two blocks A and B each have a mass of 5 kg and are...

The two blocks $A$ and $B$ each have a mass of $5 \mathrm{kg}$ and are suspended from parallel cords. A spring, having a stiffness of $k=60 \mathrm{N} / \mathrm{m},$ is attached to $B$ and is compressed $0.3 \mathrm{m}$ against $A$ and $B$ as shown. Determine the maximum angles $\theta$ and $\phi$ of the cords when the blocks are released from rest and the spring becomes unstretched.

Key Concepts

Conservation of Energy

This concept involves the transformation of energy from one form to another without any loss in the total energy of the system. In problems like this, the energy stored in the compressed spring is converted into gravitational potential energy and kinetic energy of the blocks as they swing, allowing us to relate initial energy to the maximum displacements or angles reached by the system.

Elastic Potential Energy

Elastic potential energy is the energy stored in an object when it is compressed or stretched, such as in a spring. This energy is calculated using the spring constant and the displacement from equilibrium, and it plays a crucial role in initiating motion when the spring is released.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position within a gravitational field. When the blocks swing upward after being propelled by the spring, the increase in their height results in a corresponding increase in gravitational potential energy, which is pivotal for determining the maximum angles of displacement.

Pendulum Motion and Geometric Relationships

The behavior of the suspended blocks can be modeled as pendular motion. The maximum angular displacement of the cords is determined by the relationship between the vertical displacement of the blocks, their length, and the geometry of the pendulum trajectory. This geometric analysis links the energy conversion process to specific angular measurements of the cords.

Transcript

00:02 In this problem, we have two blocks a and b, which have identical mass, and are suspended from parallel chords.

00:09 A spring with a known spring constant is attached between the blocks, and the blocks are initially at rest with the spring compressed.

00:17 We want to calculate the maximum angles of the chords when the blocks are released from rest.

00:24 So if we take block a as an example, its angle is theta when it reaches its highest point, its lowest point will regard as the datum.

00:33 The string or the cord has length to meters.

00:38 So its maximum height, when theta is in a maximum for block a is the same as the maximum height for block b at an angle of phi is 2 minus 2 cost theta.

00:52 So the first thing we will do is apply the conservation of linear momentum in the horizontal direction.

01:00 So the conservation of linear momentum tells us that the sum of the momenta before must equal to the sum of the momentum after.

01:15 Now initially both the blocks are addressed.

01:17 So they have no momentum.

01:19 So the initial momentum of the system is zero.

01:22 And the final momentum of the system, we take the right as positive, is equal to the mass of block a times its velocity after it begins to move apart.

01:36 Plus the mass of block b, which is also 5kg times its velocity vb.

01:43 So we can see that va is equal to vb, and we'll call this velocity v or the speed v.

01:52 So that's the speed at which these two blocks will begin to move apart when they are released from rest.

01:59 Now we apply the conservation of momentum.

02:02 Next we'll apply the conservation of energy.

02:05 Now the conservation of mechanical energy tells us that the initial kinetic plus gravitational potential energy of the two blocks must equal their final kinetic and gravitational potential energies.

02:20 So t1 plus v1 must equal to t2 plus v2.

02:26 Now this is just before the blocks begin to rise.

02:32 So initially we have no kinetic energy for either block.

02:44 We have no gravitational potential energy but we have potential energy stored in the spring.

02:49 That's a half kx squared.

02:52 So it's a half times 60 times the compression, which is 0 .3 meters...

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