please help me im stuck.
A 400 g air-track glider attached to a spring with spring constant 9.00 N/m is sitting at rest on a frictionless air track. A 600 g glider is pushed toward it from the far end of the track at a speed of 94.0 cm/s. It collides with and sticks to the 400 g glider.
What is the amplitude of the subsequent oscillations?
What is their period?
use conservation of momentum to find the velocity of the combined masses.
Using that, find KEinitial 1/2 (combinedmass)newvelocity^2
Set that equal to PE max 1/2 kx^2
solve for x, amplitude
Period=2PI sqrt(m/k)
To find the amplitude of the subsequent oscillations, we can use the conservation of momentum principle. Since the two gliders stick together after the collision, the total momentum before the collision is equal to the total momentum after the collision.
The total momentum before the collision can be calculated as the sum of the momenta of the individual gliders:
Total momentum before collision = (mass of first glider)*(velocity of first glider) + (mass of second glider)*(velocity of second glider)
The momentum after the collision is given by:
Total momentum after collision = (mass of combined gliders)*(velocity after collision)
Since the gliders stick together after the collision, their combined mass is the sum of their individual masses.
Now let's calculate the momentum before and after the collision:
Momentum before collision = (0.4 kg)*(0 m/s) + (0.6 kg)*(94.0 cm/s)
= 0 + (0.6 kg)*(0.94 m/s)
= 0.564 kg⋅m/s
Momentum after collision = (1 kg)*(velocity after collision)
Since the gliders stick together and move with the same velocity after the collision, the velocity after collision can be calculated using the conservation of momentum:
Momentum before collision = Momentum after collision
0.564 kg⋅m/s = (1 kg) * (velocity after collision)
Solving for velocity after collision:
velocity after collision = 0.564 kg⋅m/s / 1 kg
= 0.564 m/s
Now, let's find the amplitude of the subsequent oscillations using the relationship between the maximum displacement (amplitude) and the spring constant:
Amplitude = (mass combined)*(velocity after collision) / (spring constant)
Amplitude = ((0.4 kg + 0.6 kg) * (0.564 m/s)) / (9 N/m)
= (1 kg * 0.564 m/s) / 9 N/m
= 0.564 kg⋅m/s / 9 N/m
= 0.0627 m
Therefore, the amplitude of the subsequent oscillations is 0.0627 m.
To calculate the period, we can use the relation between the mass of the gliders and the spring constant:
Period = 2π * √(mass combined / spring constant)
Period = 2π * √((0.4 kg + 0.6 kg) / 9 N/m)
= 2π * √(1 kg / 9 N/m)
= 2π * √(1/9)
= 2π * (1/3)
= 2π/3
Therefore, the period of the subsequent oscillations is 2π/3.
To answer these questions, we need to apply the principles of conservation of momentum and conservation of energy.
First, let's consider the conservation of momentum. Before the collision, there is no external force acting on the system, so the total momentum before the collision is equal to the total momentum after the collision. Since the gliders stick together after the collision, their combined mass is 400 g + 600 g = 1000 g = 1 kg.
Using the formula for momentum (p = m * v), where p is momentum, m is mass, and v is velocity, we can write:
(m1 * v1) + (m2 * v2) = (m1 + m2) * v
Here, m1 and v1 represent the mass and velocity of the 600 g glider, and m2 and v2 represent the mass and velocity of the 400 g glider before the collision. The mass and velocity of the combined gliders after the collision are represented by (m1 + m2) and v, respectively.
Substituting the given values:
(0.6 kg * 94.0 cm/s) + (0.4 kg * 0 cm/s) = (1 kg * v)
Simplifying the equation:
56.4 kg⋅cm/s = v kg⋅m/s
Converting the units:
56.4 kg⋅cm/s = (0.0564 kg) * v m/s
v = 56.4/0.0564 = 1000 m/s
Now, let's calculate the kinetic energy of the system before the collision. The kinetic energy of an object is given by the formula K = (1/2) * m * v^2, where K is kinetic energy, m is mass, and v is velocity.
For the 600 g glider:
K1 = (1/2) * (0.6 kg) * (94.0 cm/s)^2
For the 400 g glider:
K2 = (1/2) * (0.4 kg) * (0 cm/s)^2 = 0 J
The total initial kinetic energy is given by:
K_total = K1 + K2
Next, let's consider the conservation of mechanical energy during the subsequent oscillations. When the gliders collide and stick together, the mechanical energy of the system is converted from kinetic energy to potential energy stored in the spring.
The potential energy stored in a spring is given by the formula U = (1/2) * k * x^2, where U is potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
Given that the mass of the combined gliders is 1 kg, and the spring constant is 9.00 N/m, we can equate the initial kinetic energy to the potential energy:
K_total = U
(1/2) * (0.6 kg) * (94.0 cm/s)^2 = (1/2) * (9.00 N/m) * x^2
Simplifying the equation:
(0.9 kg⋅m/s)^2 = (9.00 N/m) * x^2
Solving for x:
x^2 = [(0.9 kg⋅m/s)^2] / (9.00 N/m)
x^2 = (0.9 kg⋅m/s)^2 / (9.00 kg⋅m/s^2)
x^2 = 0.09 m^2
Taking the square root of both sides:
x = 0.3 m
The amplitude of the subsequent oscillations is therefore 0.3 m.
Finally, to calculate the period of the oscillations, we can use the formula T = 2π * √(m/k), where T is the period, m is the mass, and k is the spring constant.
Substituting the values:
T = 2π * √(1 kg / 9.00 N/m)
Simplifying the equation:
T = 2π * √(1 kg⋅m/s^2) / N
T = 2π * √(1 s^2)
T = 2π s
The period of the oscillations is equal to 2π seconds.