Two gliders with masses m1 = 0.200 kg and m2 = 0.300 kg are held stationary on an air-track with compressed spring in between. The gliders are then
released and the spring pushes them apart. If m1 moves at 3.00 m/s right after it is released, �nd the initial compression of the spring, if its spring constant is 300 N/m. Ignore the eff�ects of friction and air resistance.
To find the initial compression of the spring, we can use the conservation of momentum.
The momentum before releasing the gliders is zero since they are stationary. After releasing, the total momentum of the system should also be zero since there is no external force acting on it.
Let's denote the initial compression of the spring as x.
The momentum of m1 after it is released can be calculated using the equation:
m1 * v1 = -m2 * v2 (1)
where
m1 = mass of glider 1 = 0.200 kg
v1 = velocity of glider 1 = 3.00 m/s
m2 = mass of glider 2 = 0.300 kg
v2 = velocity of glider 2
Rearranging equation (1), we get:
v2 = -(m1 * v1) / m2
Substituting the given values, we can calculate v2:
v2 = -(0.200 kg * 3.00 m/s) / 0.300 kg
= -2.00 m/s
Now, we can calculate the change in momentum:
Δp = m2 * v2 - m2 * 0 (2)
Substituting the values, we get:
Δp = 0.300 kg * (-2.00 m/s) - 0.300 kg * 0
= -0.600 kg m/s
According to the conservation of momentum, this should be equal to the momentum gained by the spring. Since the masses are moving in opposite directions, the sign of the momentum gained by the spring will be positive:
Δp = k * x (3)
where
k = spring constant = 300 N/m
x = initial compression of the spring
Rearranging equation (3), we can solve for x:
x = Δp / k
Substituting the values, we get:
x = (-0.600 kg m/s) / (300 N/m)
= -0.002 m
The negative sign indicates that the spring was initially compressed by -0.002 m, which means it was compressed in the opposite direction to the motion of the gliders.
To solve this problem, we can use the conservation of momentum and the equation for the potential energy stored in a spring.
The conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, as long as there are no external forces acting on the system.
In this case, before the gliders are released, they are stationary. So the initial total momentum is zero.
After the gliders are released, m1 moves at 3.00 m/s to the right. Let's call the velocity of m2 v2.
The total momentum after the gliders are released can be calculated as:
Total momentum = m1 * v1 + m2 * v2
Since the initial total momentum is zero, we can set this equation equal to zero:
0 = m1 * v1 + m2 * v2
Now, let's use the equation for the potential energy stored in a spring:
Potential energy = (1/2) * k * x^2
where k is the spring constant, and x is the compression or elongation of the spring.
By releasing the gliders, the potential energy of the compressed spring is converted into the kinetic energy of the gliders. So we can equate the potential energy to the kinetic energy of m1 after it is released:
Potential energy = (1/2) * m1 * v1^2
Since we know the spring constant (k = 300 N/m) and the velocity of m1 after it is released (v1 = 3.00 m/s), we can solve for x, the initial compression of the spring, in the equation:
(1/2) * k * x^2 = (1/2) * m1 * v1^2
x^2 = (m1 * v1^2) / k
x = √((m1 * v1^2) / k)
Plugging in the values, we get:
x = √((0.200 kg * (3.00 m/s)^2) / (300 N/m))
x ≈ 0.346 m
Therefore, the initial compression of the spring is approximately 0.346 meters.