The position of a 50 g oscillating mass is given by 𝓍(t) = (2.0 cm) cos (10 t ─ π/4), where t is in s. Determine: h. The velocity at t = 0.40 s.

Verified step by step guidance

1. Identify the function for position, which is given as $x(t) = (2.0 \, \text{cm}) \cos(10t - \frac{\pi}{4})$. Recognize that this is a simple harmonic motion equation.
2. Differentiate the position function $x(t)$ with respect to time $t$ to find the velocity function $v(t)$. Use the chain rule for differentiation: $v(t) = -\sin(10t - \frac{\pi}{4}) \cdot (2.0 \, \text{cm}) \cdot 10$.
3. Simplify the expression for velocity: $v(t) = -20.0 \, \text{cm/s} \cdot \sin(10t - \frac{\pi}{4})$.
4. Substitute $t = 0.40 \, \text{s}$ into the velocity function to find the velocity at that specific time: $v(0.40) = -20.0 \, \text{cm/s} \cdot \sin(10 \times 0.40 - \frac{\pi}{4})$.
5. Calculate the value inside the sine function and then find the sine of that value to determine the velocity at $t = 0.40 \, \text{s}$. Remember to keep the units consistent.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. The motion can be described by sinusoidal functions, such as sine or cosine, which represent the position of the object as a function of time. In this case, the mass oscillates with a specific amplitude and angular frequency, which are key to determining its velocity and other properties.

Velocity in SHM

In the context of Simple Harmonic Motion, the velocity of an oscillating mass can be derived from its position function. The velocity is the time derivative of the position function, indicating how fast the mass is moving and in which direction. For the given position function, differentiating it with respect to time will yield the velocity expression, which can then be evaluated at any specific time.

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an object oscillates in radians per second. It is related to the frequency of oscillation and is a crucial parameter in the equations of motion for SHM. In the provided position function, the angular frequency is represented by the coefficient of t in the cosine term, which influences both the speed and the period of the oscillation.