Question

A ball of mass M is attached to one end of a spring of stiffness k and relaxed length L0$wVHTS+T71kGkQAAAABJRU5ErkJggg==$. The other end of the spring is attached to the ceiling. When the ball hangs at rest in equilibrium at the end of the spring it is located at the origin of the coordinate system shown and the spring’s length is Leq$O4AAAAASUVORK5CYII=$.

a. The figure shows the ball at position $KhRfqTsp+oAAAAASUVORK5CYII=$. What are the components of the vector Li that connects the point where the spring is attached to the ceiling to the ball’s position?

b. What is the magnitude of that vector, ?

c. What are the components of the net force acting on the ball if $ypk43UwAAAAASUVORK5CYII=$ kg, N/m, $cs5fENUsDi3LIoZVgAAAAASUVORK5CYII=$ m, _­$NlvvNAy14E4AAAAASUVORK5CYII=$m, and m?

d. Estimate the momentum and the position of the ball 0.1s after it is released from rest at the initial location m.

Answer 1

From the diagram, = < 0, L_eq, 0 > = < x_i, y_i, 0 > Component of vector L^bar_1 = r^bar_ - r^bar_ = < x_i, y_1 - L_eq, 0 > L^bar_i = x^2_i + (y_i - L_eq)^2 Component of net Force, F^bar_s = -K [L^bar_i - L_0] L^bar_i = -K [L^bar_i - L_0] L^bar_i /L^_i L_i = < 0.4 m, 0.9 m - 0.2 m, 0 > = < 0.4 m, 0.7 m, 0 > \r\n L^bar_i = squareroot (0.4)^2 + (0.7)^2 = 0.806 m. Spring Force F^bar_s = -[2 N/m [0.806 m - 0.4 m] < 0.496, -0.868, 0 >] = < -0.493, 0.705, 0 > F^bar_g = < 0, -mg, 0 > = < 0, -0.98, 0 > F_net = F^bar_s + F^bar_g = < -0.403, -0.275, 0 > N Momentum P_f = P^bar_i + F_net delta t = < 0, 0, 0 > + [< -0.403, -0.275, 0 > 0.1 kgm/s] = < -0.0403, -0.0275, 0 > kg m/s \r\n V_f = P_f /m = < -0.403, -0.275, 0 > position r^bar_f = r_i + v_f 4t = < 0.4, 0.2, 0 > + [< -0.403, -0.275, 0 > 0.15] =