Understanding the Velocity of an Object Undergoing Uniform Circular Motion (UCM)

Introduction to Uniform Circular Motion (UCM)

Uniform circular motion (UCM) is a type of motion where an object moves in a circular path with constant speed. Despite the constant speed, the direction of the velocity vector changes continuously. This is due to the centripetal acceleration directed towards the center of the circular path. In this article, we will derive the velocity of an object undergoing UCM using its position vector and evaluate it at a specific time.

Given Position Vector:

The position vector of an object in UCM is given by:

(mathbf{r}(t) cos{2t}mathbf{i} 0.5 sin{2t}mathbf{j})

Step 1: Calculating the Velocity Vector

To find the velocity vector, we need to take the derivative of the position vector with respect to time. This is done to determine the instantaneous rate of change of the position vector with respect to time, which represents the velocity vector.

Derivative of Position Vector:

The derivative of the position vector (mathbf{r}(t)) with respect to time (t) is:

(mathbf{v}(t) frac{dmathbf{r}(t)}{dt} -2sin{2t}mathbf{i} - "sin{2t}mathbf{j})

Here, (mathbf{i}) and (mathbf{j}) are the unit vectors in the x and y directions, respectively.

Step 2: Evaluating the Velocity at (t 2) seconds

Next, we evaluate the velocity vector at (t 2) seconds to determine the velocity at that specific time.

(mathbf{v}(2) -2sin{4}mathbf{i} - cos{4}mathbf{j})

Let's calculate the numerical values for (sin{4}) and (cos{4}).

Step 3: Numerical Calculation

Using a calculator, we find:

(sin{4} approx 0.7568)

(cos{4} approx -0.6536)

Therefore, the velocity vector at (t 2) seconds is:

(mathbf{v}(2) approx -2(0.7568)mathbf{i} - (-0.6536)mathbf{j} -1.5136mathbf{i} 0.6536mathbf{j})

Thus, the velocity vector is:

(mathbf{v}(2) approx -1.5136mathbf{i} 0.6536mathbf{j})

Additional Insights

Simplified Expression for Velocity:

The velocity vector in its simplified form is:

(mathbf{v}(2) -2sin{4}mathbf{i} cos{4}mathbf{j})

With the values, we get:

(mathbf{v}(2) -1.5136mathbf{i} 0.6536mathbf{j})

Step 4: Calculating the Speed

Speed is the magnitude of the velocity vector. To find the speed at (t 2) seconds:

(text{Speed} |mathbf{v}(2)| sqrt{(-1.5136)^2 (0.6536)^2} sqrt{2.292 0.427} sqrt{2.719} approx 1.65)

Conclusion

This example demonstrates the process of finding the velocity of an object undergoing uniform circular motion using its position vector. The velocity vector is calculated by taking the derivative of the position vector, and then evaluating it at a specific time. The results are then used to understand the direction and magnitude of the velocity at that time.

Keywords: velocity calculation, uniform circular motion, position vector derivative