Question

The ball launcher in a pinball machine has a spring that has a force constant of 1.20 N/cm (Fig. P7.63). The surface on which the ball moves is inclined 10.0

Answer 1

Force constant and compression are given in cm units , so let us convert both of them into N/m and in m respectively. Then,

Force constant =120 N/m

Compression= 0.045 m.

Mechanical energy is conserved in the above problem, which means that the potential energy in the spring must be equal to the kinetic energy of the ball on launch plus the gravitational potential energy gained by the ball as it increases in height while being pushed by the spring.

So this change in height can calculate as 0.045cos10⁰ m.

Potential energy in spring = 1/2 kx�

where k is spring constant, x is compression
Gravitational potential energy = 0.045mgsin10⁰

where m is the mass and g the acceleration due to gravity, 9.8 m/s�
Kinetic energy = 1/2 mv�

Work_spring -          Work_gravity = KE

1/2 k x_i² - 1/2 k x_f² -          mgh = KE

x_i=0

So kinetic energy is spring potential energy minus gravitational potential energy

1/2 mv� = 1/2 kx� - 0.045mgsin10. For doing the calculation easily let us multiply throughout the equation by 2 gives

mv� = kx� - 0.09mgsin10⁰.

Substituting the values gives

0.100v² = 120x0.045² - 0.09x0.100x9.8sin10⁰
0.100v²= 0.227.

Divide by 0.100

So we get v� = 2.27,

so v=1.51 m/s