Question

Four balls, each of mass m, are connected by four identical relaxed springs with spring constant k. The balls are simultaneously given equal initial speeds vdirected away from the center of symmetry of the system as shown in (Figure 1) .

a- As the balls reach their maximum displacement, their kinetic energy reaches zero why?

b- Each of the balls will move outwards to a maximum displacement d, from its initial position. Use geometry to find x, the distance each of the springs has stretched from its equilibrium position. (It may help to draw the initial and the final states of the system.)

Express your answer in terms of d.

c-

Find the maximum displacement d of any one of the balls from its initial position.

Express d in terms of some or all of the given quantities k, v, and m.

Answer 1

a. The potential energy of the ball at maximum displacement is U = 1/2kd^2. b. The kinetic energy of the ball at maximum displacement is zero. c. The maximum velocity of the ball is reached when it has zero potential energy and maximum kinetic energy.

Step-by-step explanation:

a. The potential energy of the ball at point A can be calculated using the formula U = 1/2kx^2, where k is the spring constant and x is the displacement from the equilibrium position. At maximum displacement, the ball is at point A, where x = d, and the potential energy is U = 1/2kd^2.

b. The kinetic energy of the ball at point B can be calculated using the formula KE = 1/2mv^2, where m is the mass of the ball and v is the velocity. At maximum displacement, the ball has zero velocity and therefore zero kinetic energy.

c. The maximum velocity that the ball will reach during its motion can be found using the formula PE + KE = constant. At point A, the potential energy is maximum and the kinetic energy is zero, so PE is maximum and equals the total energy of the system. Therefore, at maximum displacement, the ball will have zero potential energy and maximum kinetic energy, which means its velocity will be at its maximum.