The ball launcher in a pinball machine has a spring that has a force constant
of 1.20 N/cm (Figure 2.6). The surface on which the ball moves is inclined
10.0° with respect to the horizontal. If the spring is initially compressed 5.00
cm, find the launching speed of a 100 g ball when the plunger is released.
Friction and the mass of the plunger are negligible
To find the launching speed of the ball, we can use the concept of conservation of mechanical energy. The mechanical energy at the start (when the spring is compressed) is equal to the sum of the potential energy and the elastic potential energy stored in the compressed spring. At the end (when the ball is released), the mechanical energy is equal to the sum of the kinetic energy and potential energy.
Step 1: Calculate the potential energy of the compressed spring.
Potential energy (PE) = 1/2 * k * x^2
where k is the force constant of the spring (1.20 N/cm) and x is the compression distance (5.00 cm).
First, we need to convert the force constant from N/cm to N/m by multiplying it by 100:
k = 1.20 N/cm * 100 cm/m = 120 N/m
Now, we can calculate the potential energy:
PE = 1/2 * 120 N/m * (5.00 cm)^2 = 150 Joules
Step 2: Calculate the potential energy at the end.
Potential energy (PE) = m * g * h
where m is the mass of the ball (100 g = 0.1 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height difference due to the inclined surface.
To find h, we can use trigonometry. The height difference is the vertical displacement along the inclined surface, which can be calculated as follows:
h = x * sin(theta)
where x is the compression distance (5.00 cm) and theta is the angle of inclination (10.0°).
First, convert the compression distance from cm to meters:
x = 5.00 cm * 0.01 m/cm = 0.05 m
Now, we can calculate the height difference:
h = 0.05 m * sin(10.0°) = 0.00872 m
Next, we can calculate the potential energy:
PE = 0.1 kg * 9.8 m/s^2 * 0.00872 m = 0.00854 Joules
Step 3: Apply the law of conservation of mechanical energy.
The mechanical energy at the start (potential energy of the compressed spring) is equal to the sum of the mechanical energy at the end (potential energy and kinetic energy of the ball).
Potential energy at the start = Potential energy at the end
PE = 0.00854 J + 0.5 * m * v^2
where v is the launching speed we want to find.
Rearranging the equation to solve for v:
0.5 * m * v^2 = PE - 0.00854 J
v^2 = (PE - 0.00854 J) * 2 / m
v = sqrt[(PE - 0.00854 J) * 2 / m]
Now, substitute the values we've already calculated and solve for v:
v = sqrt[(150 J - 0.00854 J) * 2 / 0.1 kg]
v = sqrt[299.991 J / 0.1 kg]
v = sqrt[2999.91 m^2/s^2 / kg]
v = sqrt[2999.91 m^2/s^2] (kg cancels out)
v ≈ 54.829 m/s
Therefore, the launching speed of the 100 g ball when the plunger is released is approximately 54.829 m/s.