Question

The position of a body oscillating on a spring is given by x = A sin wt, where A and v are constants with values A = 5 cm and w = 0.175 s^-1. (a) Sketch x versus t for 0 less than or equal to t less than or equal to 36 s. (b) Measure the slope of your graph at t = 0 to find the velocity at this time. (c) Calculate the average velocity for a series of intervals beginning at t = 0 and ending at t = 6, 3, 2, 1, 0.5, and 0.25 s. (d) Compute dx/dt and find the velocity at time t = 0. 

Answer 1

a)

$$x=5\sin{0.175t}$$

b)

$$v=0.875\frac{cm}{s}$$

c)

$$v_{av}=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}vdt=\frac{x_2-x_1}{t_2-t_1}$$

$$v_1=\frac{5\sin{0.175(6)}}{6}=0.723\frac{cm}{s}\\v_2=\frac{5\sin{0.175(3)}}{3}=0.835\frac{cm}{s}\\v_3=\frac{5\sin{0.175(2)}}{2}=0.857\frac{cm}{s}\\ v_4=\frac{5\sin{0.175(1)}}{1}=0.871\frac{cm}{s}\\v_5=\frac{5\sin{0.175(0.5)}}{0.5}=0.874\frac{cm}{s}\\v_6=\frac{5\sin{0.175(0.25)}}{0.25}=0.875\frac{cm}{s}\\$$

d)

$$v=0.875\frac{cm}{s}$$