Problem 58 - College Physics

Question

Spiraling up. It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00 \(\mathrm{m}\) every 5.00 \(\mathrm{s}\) and rises vertically at a rate of 3.00 \(\mathrm{m} / \mathrm{s} .\) Determine: (a) the speed of the birc relative to the ground; (b) the bird's acceleration (magnitude) and direction); and (c) the angle between the bird's velocity vector and the horizontal.

Step-by-Step Solution

Answer

(a) 10.49 m/s, (b) 12.63 m/s² towards circle's center, (c) 16.70° above the horizontal.

Step 1: Find the Circular Speed

Calculate the circular speed of the bird by using the formula for the circumference of a circle and the time taken to complete one circle. The circumference is given by \( C = 2\pi r \), where \( r = 8.00 \, \text{m} \). Thus, \( C = 2 \pi \times 8.00 = 16\pi \). Since the bird completes this circle every 5.00 seconds, the speed is \( v = \frac{C}{T} = \frac{16\pi}{5.00} \approx 10.05 \, \text{m/s} \).

Step 2: Determine Total Speed

The bird has a circular speed from Step 1, and an upward vertical speed of 3.00 m/s. To find the total speed, use the Pythagorean theorem because these velocities are perpendicular: \( v_{\text{total}} = \sqrt{(v_{\text{circular}})^2 + (v_{\text{vertical}})^2} = \sqrt{(10.05)^2 + (3.00)^2} \approx 10.49 \, \text{m/s} \).

Step 3: Calculate Acceleration

The bird's acceleration is solely due to the circular motion; this is the centripetal acceleration. Use the formula \( a = \frac{v_{\text{circular}}^2}{r} = \frac{(10.05)^2}{8.00} \approx 12.63 \, \text{m/s}^2 \). This acceleration is directed towards the center of the circle.

Step 4: Find the Angle of Velocity Vector

The angle \( \theta \) between the bird's velocity vector and the horizontal can be found using the tangent function: \( \tan \theta = \frac{v_{\text{vertical}}}{v_{\text{circular}}} = \frac{3.00}{10.05} \). Thus, \( \theta = \tan^{-1}\left(\frac{3.00}{10.05}\right) \approx 16.70^\circ \).

Key Concepts

Centripetal Acceleration

When an object moves in a circular path, it is constantly changing direction. Even though an object may have a constant speed, the change in direction means there is an acceleration present. This acceleration is known as centripetal acceleration. It is always directed towards the center of the circle the object is moving around.

The formula to calculate centripetal acceleration (\( a \)) is given by:

• \( a = \frac{v^2}{r} \)

where \( v \) is the circular speed and \( r \) is the radius of the circle.

In our example, the bird experiences this type of acceleration as it spirals upward. With a circular speed of 10.05 m/s and a radius of 8.00 m, the centripetal acceleration calculates to approximately 12.63 m/s². This value tells us how swiftly the bird must change its velocity vector to maintain its circular path.

Velocity Vector

In the realm of physics, a velocity vector is an essential concept that combines both the speed and direction of an object's movement. Essentially, velocity is a vector quantity, which means it has both magnitude and direction. For our bird spiraling upward, the velocity vector is a combination of the horizontal circular movement and the vertical ascent.

To find the bird’s overall speed, we combine the circular speed (10.05 m/s) and the vertical speed (3.00 m/s) using the Pythagorean theorem. This gives us the total speed or the magnitude of the velocity vector, approximately 10.49 m/s.

The direction of this velocity vector is crucial to understanding how the bird moves. Knowing both the horizontal circular and upward vertical components helps us paint a complete picture of the bird's dynamic spiral flight.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used in geometry that relates the lengths of the sides of a right triangle. It states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:

• \( c^2 = a^2 + b^2 \)

where \( c \) is the hypotenuse and \( a \) and \( b \) are the two other sides of the triangle.

In the context of our spiraling bird, the Pythagorean theorem helps calculate the total speed. The bird moves in a circular path (10.05 m/s) horizontally, and rises vertically (3.00 m/s). Using the theorem, the total speed (hypotenuse) of the bird is:

• \( v_{\text{total}} = \sqrt{10.05^2 + 3.00^2} \approx 10.49 \, \text{m/s} \)

This application of the Pythagorean theorem elegantly connects geometric principles to physical movement.

Tangent Function

The tangent function is a concept from trigonometry that helps relate angles to the sides of a right triangle. In terms of circular motion and spiraling paths, the tangent function is instrumental in determining the angle of a velocity vector with respect to a reference line, such as the horizontal.

For the bird in our example, its velocity vector forms an angle with the horizontal plane due to its vertical and circular speeds. This angle \( \theta \) is found using:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{v_{\text{vertical}}}{v_{\text{circular}}} \)

Here, \( v_{\text{vertical}} \) is 3.00 m/s and \( v_{\text{circular}} \) is 10.05 m/s, giving:

• \( \theta = \tan^{-1}\left(\frac{3.00}{10.05}\right) \approx 16.70^\circ \)

This angle demonstrates the efficient use of the tangent function to decipher the bird's movement path.