The figure below shows a block of mass m (Block 1) that is...

The figure below shows a block of mass m (Block 1) that is attached to one end of an ideal spring of force constant k and natural length L. The block is pushed so that it compresses the spring to 3/4 of its natural length and is then released from rest. Just as the spring has extended to its natural length L, the attached block collides with another block (also of mass m) at rest on the edge of the frictionless table. When Block 1 collides with Block 2, half of its kinetic energy is lost to heat; the other half of Block 1’s kinetic energy at impact is divided between Block 1 and Block 2. The collision sends Block 2 over the edge of the table, where it falls a vertical distance H, landing at a horizontal distance R from the edge. (a) What is the acceleration of Block 1 at the moment it's released from rest from its initial position? Write your answer in terms of $k, L,$ and $m$ . (b) If $v_{1}$ is the velocity of Block 1 just before impact, show that the velocity of Block 1 just after impact is $\frac{1}{2} v_{1}$ . (c) Determine the amplitude of the oscillations of Block 1 after Block 2 has left the table. Write your answer in terms of $L$ only. (d) Determine the period of the oscillations of Block 1 after the collision, writing your answer in terms of $T_{0},$ the period of the oscillations that Block 1 would have had if it did not collide with Block 2 . (e) Find an expression for $R$ in terms of $H, k, L, m,$ and $g$ .

The figure below shows a block of mass m (Block 1) that is attached to one end of an ideal spring of force constant k and natural length L. The block is pushed so that it compresses the spring to 3/4 of its natural length and is then released from rest. Just as the spring has extended to its natural length L, the attached block collides with another block (also of mass m) at rest on the edge of the frictionless table. When Block 1 collides with Block 2, half of its kinetic energy is lost to heat; the other half of Block 1’s kinetic energy at impact is divided between Block 1 and Block 2. The collision sends Block 2
over the edge of the table, where it falls a vertical distance H, landing at a horizontal distance R from the edge.
(a) What is the acceleration of Block 1 at the moment it's released from rest from its initial position? Write your answer in terms of $k, L,$ and $m$ .
(b) If $v_{1}$ is the velocity of Block 1 just before impact, show that the velocity of Block 1 just after impact is $\frac{1}{2} v_{1}$ .
(c) Determine the amplitude of the oscillations of Block 1 after Block 2 has left the table. Write your answer in terms of $L$ only.
(d) Determine the period of the oscillations of Block 1 after the collision, writing your answer in terms of $T_{0},$ the period of the oscillations that Block 1 would have had if it did not collide with Block 2 .
(e) Find an expression for $R$ in terms of $H, k, L, m,$ and $g$ .

Key Concepts

Hooke’s Law

Hooke’s Law establishes that the force exerted by a spring is directly proportional to its displacement from its equilibrium (natural) length. This concept is essential for determining the restoring force and the resulting acceleration of a mass attached to a spring, as well as for calculating the potential energy stored in the spring during compression or extension.

Simple Harmonic Motion

Simple Harmonic Motion refers to the periodic oscillatory movement of a system where the restoring force is proportional to the displacement from equilibrium. This concept is critical for understanding the behavior of oscillatory systems, such as the amplitude and period of a mass attached to a spring, and it informs how changes to initial conditions (like post-collision energy) affect the oscillation parameters.

Conservation of Energy

Conservation of Energy is the principle that the total energy in an isolated system remains constant, though it may transform between different forms. In problems involving spring motion, this concept is used to track the conversion of elastic potential energy in the compressed spring into kinetic energy of the moving block, as well as to analyze energy changes during collisions.

Inelastic Collision

An inelastic collision is one in which kinetic energy is not conserved due to energy losses (often converted into heat, sound, or deformation), even though momentum is conserved. This concept is crucial when analyzing collisions where a part of the kinetic energy is lost, influencing the post-collision speeds of the colliding objects.

Conservation of Momentum

Conservation of Momentum states that the total momentum of an isolated system remains unchanged in the absence of external forces. This principle is employed in collision problems to determine the velocities of objects after they interact, regardless of whether the collision is elastic or inelastic.

Projectile Motion

Projectile Motion describes the motion of an object that is launched into the air and is under the influence of gravitational acceleration. It is used to analyze trajectories, determine horizontal distances traveled, and calculate the time of flight when an object moves off a table or similar platform and falls under gravity.

Transcript

00:01 So this problem, we have the setup shown as here.

00:06 And in part a, we want to, let me see.

00:14 So in part a, we want to find an acceleration of block a.

00:16 At the moment, it is released from the rest, from the initial position.

00:21 So we know that it is compressed by 1 over 4 times l, right? so according to the hoax law, the force acting on the block is 1 over 4 times l times k.

00:34 This equal m times a, right? so the initial acceleration is kl over full m.

00:43 Okay.

00:44 And then b says that we want to, so if the v1 is the velocity of block 1, before impact, show that the velocity of block 1 just after impact is half v1.

01:02 So because according to a problem, we know that only half of the kinetic energy is transferred into block 2, right? so we see that, i'm sorry, half of the kinetic energy is transferred into heat.

01:19 So only half of the kinetic energy is transferred into the mechanical energy, right? so we have half times half mv1 squared.

01:30 So this means half of the initial kinetic energy equal half mv1 prime squid plus half mv2 prime squid.

01:39 Excuse me so v1 v2 means the velocity of a block 1 the block 2 after the collision okay so this is the this is the energy conservation law and then we have the momentum reservation law we have m times v1 equal m times a v1 prime plus m times v2 prime right so we have this two we have this two equations and only an unknown variable is the v1 prime the v2 prime so if you solve this two equations, you will obtain that to v1 prime equal v2 prime, equal half times v1, all right.

02:20 And in part c, we want to show that to the oscillation amplitude after the collision...

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