a body oscillates with simple harmonic motion along the x axis. Its displacement varies with time according to the equation x = 5.0 cos (pt). The magnitude of the acceleration (in m/s2) of the body at t = 1.0 s is approximately
a. 3.5 b.14 c.43 d.4.3
14
43
To find the acceleration of the body, we need to take the second derivative of the displacement equation.
Given: x = 5.0 cos(pt)
Taking the derivative once with respect to time (t), we get:
v = dx/dt = -5.0p sin(pt)
Taking the derivative again with respect to time (t), we get:
a = dv/dt = d²x/dt² = -5.0p² cos(pt)
Now, we can substitute the value of t = 1.0 s into the equation for acceleration:
a = -5.0p² cos(p * 1.0)
Since the value of p is not given, we cannot determine the exact value of acceleration. However, we can approximate it by using the value of pi (π) as 3.14.
So, substituting p = 3.14 into the equation, we get:
a ≈ -5.0 * (3.14)² * cos(3.14 * 1.0)
a ≈ -5.0 * 9.8596 * cos(3.14)
a ≈ -49.298 * cos(3.14)
a ≈ -49.298 * (-1)
a ≈ 49.298
Therefore, the approximate magnitude of the acceleration of the body at t = 1.0 s is approximately 49.3 m/s².
None of the given options (a, b, c, or d) match this value.
To find the magnitude of acceleration at t = 1.0 s, we need to differentiate the given displacement equation twice with respect to time.
The given displacement equation is x = 5.0 cos(pt).
Differentiating once, we get the velocity equation:
v = dx/dt = -5.0p sin(pt).
Next, let's differentiate the velocity equation with respect to time:
a = dv/dt = d²x/dt² = -5.0p² cos(pt).
Now, we can substitute t = 1.0 s into the acceleration equation to find the magnitude of acceleration at t = 1.0 s:
a = -5.0p² cos(p * 1.0).
Since p is not specified, we cannot calculate the exact value. However, we can determine the approximate magnitude by considering the value of cos(p * 1.0).
The cosine function has a maximum value of 1 when the angle is 0 degrees (or multiples of 360 degrees). Therefore, the maximum value for cos(p * 1.0) is 1.
So, the magnitude of acceleration at t = 1.0 s is approximately:
a ≈ -5.0p² * 1 ≈ -5.0p².
Since we do not have the value of p, we cannot calculate the exact magnitude. However, based on the given options, the closest approximation is 4.3 (option d).