A 0.490-kg block is attached to a horizontal spring that is at its equilibrium length, and whose force constant is 25.0N/m . The block rests on a frictionless surface. A 6.00×10−2-kg wad of putty is thrown horizontally at the block, hitting it with a speed of 2.10m/s and sticking.
How far does the putty-block system compress the spring? (In other words what is X max)
To determine how far the putty-block system compresses the spring, we can use the principle of conservation of linear momentum.
The initial momentum of the putty is given by the formula p_initial = m_putty * v_putty, where m_putty is the mass of the putty and v_putty is its initial velocity.
The final momentum of the putty and block system can be calculated using the formula p_final = (m_putty + m_block) * v_final, where m_block is the mass of the block and v_final is the final velocity of the putty and block system.
Since the putty sticks to the block after the collision, the final velocity of the putty-block system can be determined using the equation of conservation of linear momentum:
p_initial = p_final
m_putty * v_putty = (m_putty + m_block) * v_final
Now, rearranging the equation to solve for v_final:
v_final = (m_putty * v_putty) / (m_putty + m_block)
Substituting the given values:
v_final = (0.06 kg * 2.10 m/s) / (0.06 kg + 0.49 kg)
v_final = 0.126 / 0.55
v_final = 0.229 m/s
The maximum compression of the spring, denoted as X_max, can be calculated using Hooke's Law. Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The equation can be written as:
F = -k * x
where F is the force exerted by the spring, k is the force constant, and x is the displacement from the equilibrium position.
Since the spring is initially at its equilibrium length, the initial force F_initial is zero. Thus, we can write:
F_final = -k * x_max
Considering Newton's second law:
F_final = m_block * a
where a is the acceleration of the block.
We know that acceleration a is related to the displacement x_max by the equation:
a = -omega^2 * x_max
where omega (ω) is the angular frequency of the spring, given by:
omega = sqrt(k / m_block)
Substituting the given values:
omega = sqrt(25 N/m / 0.49 kg)
omega = 3.19 rad/s
Now we can calculate the force exerted by the spring using:
F_final = -k * x_max
m_block * a = -k * x_max
m_block * (-omega^2 * x_max) = -k * x_max
Rearranging the equation to solve for x_max:
x_max = -m_block * omega^2 / k
Substituting the given values:
x_max = -0.49 kg * (3.19 rad/s)^2 / 25 N/m
x_max = -0.49 * 3.19^2 / 25
x_max = -0.49 * 10.1761 / 25
x_max = -0.490 / 2.5
x_max = -0.196 m
Therefore, the putty-block system compresses the spring by approximately 0.196 meters (or 19.6 cm).