Principle of conservation of linear momentum equation

[SirPlus]

Homework Statement

A massless spring attached to a wall lies on a frictionless table. It has a block of mass 2kg attached to one end, initially the block is at rest. Another block, also of mass 2kg is sliding on the table top with a speed 8m/s. At t = o the moving block collides with the block on the spring . The two stick together and oscillate back and forth. If the spring constant is 16 n/m, find an expression x(t) which describes the motion of the blocks that are stuck together.

Homework Equations

-Principle of conservation of linear momentum equation

The Attempt at a Solution

Step 1:
I applied principle of conservation of linear momentum to the system to find the velocity after the collision.

Step 2:
I obtained the angular velocity with the data provided

Step 3:
I associated the harmonic motion of the system to that of a sin function - the rest/equilibrium postion equals zero

Step 4: I don't know how to obtain the maximum positions for the oscillation -

In my attempt the position at any time of the SHM is x(t) = 2 * sin2t

Please correct me if i am wrong ...

 

[siddharth23]

I don't remember the formulae, but use F=kx for the spring. Use F=ma (so basically kx=ma) Acceleration in SHM is opposite to the direction of travel Using that, you'll get an euqation for your SHM.

The first 3 points are correct. The point of impact of the two blocks is the equilibrium position.

 

[ehild]

Homework Statement

A massless spring attached to a wall lies on a frictionless table. It has a block of mass 2kg attached to one end, initially the block is at rest. Another block, also of mass 2kg is sliding on the table top with a speed 8m/s. At t = o the moving block collides with the block on the spring . The two stick together and oscillate back and forth. If the spring constant is 16 n/m, find an expression x(t) which describes the motion of the blocks that are stuck together.

Homework Equations

-Principle of conservation of linear momentum equation

The Attempt at a Solution

Step 1:
I applied principle of conservation of linear momentum to the system to find the velocity after the collision.

Step 2:
I obtained the angular velocity with the data provided

Step 3:
I associated the harmonic motion of the system to that of a sin function - the rest/equilibrium postion equals zero

Step 4: I don't know how to obtain the maximum positions for the oscillation -

But you know the maximum speed...

In my attempt the position at any time of the SHM is x(t) = 2 * sin2t

Please correct me if i am wrong ...

How did you get it? And how did you define x? In what direction is it positive? In what units is it written?

Spoiler

(If you specify the direction and units, it is correct)

ehild

 

[SirPlus]

I obtained the maximum speed using the principle of conservation of linear momentum, i took the first derivative of the postion function with respect to time and equated the initial velocity at time zero - i then was able to determine the maximum displacement. Direction positive is along the positive x - axis(to the right).

I just needed you to verify whether my approache is OK

 

[Nickie]

Your approach to solving this problem is correct so far. The principle of conservation of linear momentum states that the total momentum of a closed system remains constant. In this case, the closed system is the two blocks stuck together after the collision. Before the collision, the momentum of the system is 2kg * 8m/s = 16kgm/s. After the collision, the momentum is still 16kgm/s, but it is now shared between the two blocks, which are moving together as one unit. Therefore, the velocity of the combined blocks after the collision is 8m/s.

To find the expression for the position of the blocks, we can use the equation for simple harmonic motion, which is x(t) = A * cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase angle. In this case, the amplitude A can be found by considering the maximum displacement of the blocks from their equilibrium position, which is the length of the spring when it is at rest. This can be calculated using Hooke's law: F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the maximum force exerted by the spring is 16N (since the mass of the blocks is 2kg and the spring constant is 16N/m), and the maximum displacement is the length of the spring, which is unknown. Therefore, A = F/k = 16N/16N/m = 1m.

Next, we can find the angular frequency ω using the equation ω = √(k/m), where k is the spring constant and m is the mass of the combined blocks. In this case, the mass is 4kg (2kg + 2kg), so ω = √(16N/m/4kg) = 2 rad/s.

Finally, the phase angle φ can be found by considering the initial conditions of the system. At t=0, the blocks are at their equilibrium position, so x(0) = 0. This means that cos(φ) = 0, which implies that φ = π/2.

Putting all of this together, we get the expression for the position of the blocks as x(t) = 1m * cos(2t +