Question

The position of a 55 g oscillating mass is given by x(t)=(2.0cm)cos(10t), where t is in seconds. determine the velocity at t=0.40s

Answer 1

For this problem, we combine the concepts learned in physics and in calculus. The velocity, by definition, is the total distance travelled per time elapsed. It can also be expressed in Δx/Δt, This is also a definition in calculus where dx/dt is equal to velocity. Therefore, to solve the velocity, differentiate the equation in terms of t.

x = 2 cos(10t)
dx/dt = 2*(-sin(10t))*(10)
dx/dt = -20sin (10t)

We are asked to find the velocity at 0.40 seconds. Thus, we substitute t = 0.40 to the equation

dx/dt = -20sin(10*0.4)
dx/dt = v = -1.395 m/s

Therefore, the velocity at t=0.04 seconds is -1.395 m/s. The negative sign connotes that the direction of the motion is south or to the left based on the sign convention.

Answer 2

The velocity of the oscillating mass at time $t = 0.40\,{\text{s}}$ is $\boxed{15\times{10^{-2}}\,{\text{m/s}}}$ or $\boxed{15\,{\text{cm/s}}}$ or $\boxed{0.15\,{\text{m/s}}}$.

Further explanation:

Velocity of a particle or a mass at any instant is defined as the rate of change of position of particle with respect to time.  

Mathematically,

$V\left( t \right) = \dfrac{d}{{dt}}\left( {X\left( t \right)} \right)$

If position of a particle or mass is a function of time then velocity of mass at any instant will change with respect to time.

Given:

The position of an oscillating mass varies according to the function  $X(t)=({2.0\text{ cm}}})\cos({10t})$.

Mass of an oscillating object is $55\text{ g}$.

Concept:

The velocity of mass at any instant is calculated by using the following relation

$\begin{aligned}V(t)&=\frac{{dX\left( t \right)}}{{dt}}\\&=\frac{d}{{dt}}\left[{\left( {2.0{\kern 1pt} {\text{cm}}} \right)\cos \left( {10t} \right)} \right]\\&=-\left( {20\,{\text{cm/s}}}\right)\sin\left( {10t}\right)\\\end{aligned}$

Therefore the velocity of the mass at any instant is given by

$V\left( t \right)=-\left( {20{\kern 1pt} {\text{cm/s}}} \right)\sin \left( {10t} \right)$                                                                          

From the above expression of velocity it can be observed that velocity is changing with time according to the sin function.

Substitute $0.40\,{\text{s}}$ for t in the above expression

$\begin{aligned}V\left( {0.4\,{\text{s}}} \right)&=-\left( {20\,{\text{cm/s}}} \right)\sin \left( 4 \right)\\&=-\left( {20\,{\text{cm/s}}} \right)\left( { - 0.757} \right)\\&=15.14\,{\text{cm/s}}\\\end{aligned}$

 

Thus, the velocity of the oscillating mass at time $t = 0.40\,{\text{s}}$ is $\boxed{15\times{10^{-2}}\,{\text{m/s}}}$ or $\boxed{15\,{\text{cm/s}}}$ or $\boxed{0.15\,{\text{m/s}}}$.

Learn More:

1.  The units expressing the rotational velocity https://brainly.com/question/2887706

2.  Find the vertical displacement as a function of time https://brainly.com/question/2887706

3. A remote controlled car is moving in a vacant parking lot https://brainly.com/question/2005478

Answer Details:

Grade: College

Subject: Physics

Chapter: Force

Keywords:

position, oscillating, 55 g, mass, time, x(t)=(2.0cm)cos(10t), t=4 s, determine, velocity, 15 cm/s, 0.15 m/s, rate, change in position.