Question

The position of a mass oscillating on a spring is given by x = ( 3.6 cm)cos[2pi t/(0.67s)].

A. What is the period of this motion?
T=? s
B. What is the first time the mass is at the position x = 0?
t=? s

Answer 1

The period of the mass oscillating on a spring is 0.67 seconds. The first time the mass is at the position x = 0 is at 0.1675 seconds.

Step-by-step explanation:

The period T of the oscillatory motion described by the equation x = ( 3.6 cm)cos[2pi t/(0.67s)] can be determined by looking at the argument of the cosine function. The period T is given by the time it takes for the argument of the cosine to increase by 2π, which corresponds to one full cycle of the oscillation. In the given equation, the factor (2π/0.67s) is multiplied by t, indicating that the period T is 0.67 seconds.

The first time the mass is at the position x = 0 after being released from rest can be found by setting the argument of the cosine to an angle where cosine is zero, such as π/2 or 3π/2. As cosine has a value of zero at these points, and the first instance this happens is at π/2, we solve for t using the equation 2πt/T = π/2. Substituting T = 0.67s into the equation gives us the first time t which is 0.1675 seconds.

Answer 2

The period of the motion is 0.67/2π seconds. The first times the mass is at the position x = 0 are t = 0 and t = T/2.

Step-by-step explanation:

The period of the motion can be determined by looking at the coefficient of t in the argument of the cosine function. In this case, the argument is 2πt/0.67s. The coefficient of t is 2π/0.67, and the period T is the reciprocal of this coefficient. Therefore, T = 0.67/2π seconds.

To find the first time the mass is at the position x = 0, we need to solve the equation x = 0. Substituting x = 0 into the expression x = (3.6 cm)cos[2πt/(0.67s)], we get 0 = (3.6 cm)cos[2πt/(0.67s)]. Solving for t, we find that t = 0 and t = T/2 are the times when the mass is at the position x = 0. Using the period T calculated earlier, we can find t = 0 and t = T/2 as the first times the mass is at the position x = 0.