A 0.57 kg block oscillates back and forth along a straight line on a frictionless horizontal surface. Its displacement from the origin is given by
x = (13 cm)cos[(20 rad/s)t + π/2 rad]
(a) What is the oscillation frequency (in Hz)? (b) What is the maximum speed acquired by the block? (c) At what value of x does this occur? (d) What is the magnitude of the maximum acceleration of the block? (e) At what positive value of x does this occur? (f) What force, applied to the block by the spring, results in the given oscillation?
How do you even start this problem? Please help
first find the period. That is when the argument of the cos changes by 2 pi
when t = 0
20(0) + pi/2 = pi/2
2 pi + Pi/2 = 2 pi + pi/2
so
period when 20t = 2 pi
or T = 2 pi/20 = pi/10
f = 1/T = 10/pi
b
dx/dt = .13 (20)(-sin(20t+pi/2))
max when |sin| = 1
dx/dt max = 2.6 m/s
c
well at lots of values of x but one is when
20t+pi/2 = pi/2
or t = 0
d take the next derivative and repeat
e now you have the acceleration d^2x/dt^2
F = m a
To solve this problem, we need to break it down into smaller steps. Let's go through each part of the problem step by step:
(a) To find the oscillation frequency, we can use the formula f = ω / (2π), where ω is the angular frequency given in the equation. In this case, ω = 20 rad/s. Therefore, the oscillation frequency is f = 20 / (2π) ≈ 3.18 Hz.
(b) The maximum speed acquired by the block occurs when the displacement x is at its maximum. We can determine the maximum displacement by looking at the coefficient of the cosine function. In this case, the maximum displacement is 13 cm. To find the maximum speed, we need to take the derivative of the displacement equation with respect to time, which gives us the velocity equation. Then, we can substitute the maximum displacement of 13 cm into the velocity equation to find the maximum speed.
(c) To find the value of x at which the maximum speed occurs, we need to substitute the expression for maximum displacement in the displacement equation and solve for t.
(d) The maximum acceleration of the block occurs when the displacement x is at its maximum or minimum. We can calculate the acceleration by taking the second derivative of the displacement equation with respect to time.
(e) To find the positive value of x at which the maximum acceleration occurs, we need to substitute the expression for the maximum or minimum value of x in the displacement equation and solve for t.
(f) To determine the force applied to the block by the spring, we need to use Hooke's law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. In this case, the displacement equation represents the motion of the block attached to a spring. So, we can infer that the force applied by the spring will be proportional to the displacement x given in the equation.