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).