A launching mechanism consists of an arm (uniform rod with mass 250. g and length 75.0 cm) with a holder (negligible mass) at one end mounted to a hub of radius 3.00 cm and negligible mass at the other end. A string is wrapped around the hub and passes to a spring of constant k = 2550 N/m that is unstretched when the launcher is in the vertical position. The launcher arm is pulled into the horizontal position and loaded with a 50.0 g point mass. It is released from rest and stops by hitting a pin at the vertical position. The projectile leaves the launcher with an initial horizontal velocity at a height of 2.00 m above the ground. Where does it land?

To determine where the projectile launched from the mechanism will land, we need to analyze the components of its motion.

1. First, let's calculate the potential energy stored in the spring when the arm is pulled into the horizontal position. The gravitational potential energy of the point mass is given by:

\(PE_{\text{gravity}} = mgh\)

where:
m = mass of the point mass = 50.0 g = 0.050 kg
g = acceleration due to gravity = 9.8 m/s^2
h = vertical distance above the ground = 2.00 m

So, \(PE_{\text{gravity}} = 0.050 \times 9.8 \times 2.00\) J

2. We also need to calculate the potential energy stored in the spring when the arm is released. The spring potential energy is given by:

\(PE_{\text{spring}} = 0.5kx^2\)

where:
k = spring constant = 2550 N/m
x = compression/elongation of the spring from its unstretched position

Note that we need to determine the elongation/compression of the spring. We can use the relation between the elongation and the angular displacement of the arm.

3. The angular displacement of the arm can be calculated using the equation:

\(L = I \cdot \theta\)

where:
L = angular momentum = mvr [by conservation of angular momentum]
m = mass of the point mass = 50.0 g = 0.050 kg
v = velocity of the point mass
r = distance of the point mass from the axis of rotation (hub radius) = 3.00 cm = 0.03 m

Since the point mass is released from rest and there is no external torque acting on it, the angular momentum is conserved.

4. Solving for the velocity of the point mass:

\(L = mvr \Rightarrow v = \frac{L}{mr}\)

Substituting the values, we have:

\(v = \frac{0.050 \times 0.03}{0.050 \times 0.03}\) m/s

This gives us the velocity of the point mass when it leaves the launcher.

5. Now, we can calculate the elongation/compression of the spring using the equation:

\(F_{\text{spring}} = kx\)

where:
F_spring = restoring force from the spring when the arm is released (equal to the weight)
x = elongation/compression of the spring

The weight is given by:

\(F_{\text{gravity}} = mg\)

where:
m = mass of the point mass = 50.0 g = 0.050 kg
g = acceleration due to gravity = 9.8 m/s^2

The restoring force of the spring is equal to the weight of the point mass. Therefore:

\(F_{\text{spring}} = F_{\text{gravity}} \Rightarrow kx = mg\)

Substituting the values, we can solve for x:

\(2550x = 0.050 \times 9.8\) N

6. Now that we have the elongation/compression of the spring, we can calculate the potential energy stored in the spring:

\(PE_{\text{spring}} = 0.5kx^2\)

Substituting the values, we can calculate the spring potential energy.

7. The total mechanical energy of the system is conserved, neglecting the effects of air resistance. Therefore:

\(PE_{\text{gravity}} + PE_{\text{spring}} = KE\) [conservation of mechanical energy]

where KE is the kinetic energy of the point mass when it leaves the launcher.

8. We can find KE using the equation:

\(KE = \frac{1}{2}mv^2\) [(1/2)mv^2 expression for kinetic energy]

Substituting the known values, we can calculate KE.

9. Now we have the kinetic energy of the point mass when it leaves the launcher. We can use this to determine the range (horizontal distance traveled) of the projectile.

The formula for the horizontal range is:

\(R = \frac{v^2}{g}\) [horizontal range formula]

where:
R = range
v = horizontal component of velocity when the projectile leaves the launcher
g = acceleration due to gravity

Substituting the values, we can calculate the horizontal range (R).