Problem 46 - University Physics with Modern Physics

Question

Birds of prey typically rise upward on thermals. The paths these birds take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume that a bird completes a circle of radius 6.00 m every 5.00 s and rises vertically at a constant rate of 3.00 m/s. Determine (a) the bird's speed 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) Speed: 8.15 m/s; (b) Acceleration: 9.48 m/s², centripetal; (c) Angle: 21.8°.

Step 1: Determine the Circular Speed

First, calculate the speed of the bird as it moves in a circular path. The speed of an object in uniform circular motion is given by \( v_c = \frac{2\pi r}{T} \), where \( r \) is the radius and \( T \) is the period of motion. For this problem, \( r = 6.00 \) m and \( T = 5.00 \) s. Thus, \( v_c = \frac{2\pi \times 6.00}{5.00} \approx 7.54 \) m/s.

Step 2: Calculate the Total Speed Relative to the Ground

The bird's overall speed relative to the ground includes both its upward velocity and its circular speed. Given the upward velocity \( v_u = 3.00 \) m/s, use the Pythagorean theorem to find the total speed: \( v = \sqrt{v_c^2 + v_u^2} = \sqrt{(7.54)^2 + (3.00)^2} \approx 8.15 \) m/s.

Step 3: Determine the Bird's Acceleration

For uniform circular motion, the centripetal acceleration, which points towards the center of the circle, is calculated by \( a_c = \frac{v_c^2}{r} \). Substituting the values, we get \( a_c = \frac{(7.54)^2}{6.00} \approx 9.48 \) m/s².

Step 4: Calculate the Angle with the Horizontal

The angle \( \theta \) between the velocity vector and the horizontal can be found using \( \tan \theta = \frac{v_u}{v_c} \). Thus, \( \theta = \tan^{-1} \left( \frac{3.00}{7.54} \right) \approx 21.8^\circ \).

Key Concepts

Spiral Motion

Birds of prey often engage in spiral motion as they ascend in the sky using thermals, or warm air currents. This allows them to rise with minimal energy expenditure. The spiral motion can be visualized as a combination of circular motion along a horizontal path, with a constant upward movement.

The key attributes of spiral motion include:

• **Circular Component**: The bird moves in a circle of radius 6.00 meters, completing one revolution every 5.00 seconds. This is a perfect example of uniform circular motion, where the speed around the path remains constant.
• **Vertical Ascent**: The bird additionally moves upward at a steady rate of 3.00 meters per second, which adds a vertical component to its overall motion.
• **Resulting Motion**: When combined, these motions create a spiral path that rises upward, characterized by both rotational and linear movement simultaneously.

Understanding spiral motion helps us get a clear picture of how complex movements can be reduced to simpler, simultaneous motions. This is particularly useful in physics for dissecting compounded vectors into comprehensible parts.

Centripetal Acceleration

Centripetal acceleration is a crucial concept when an object moves in a circular path. It is defined as the acceleration directed towards the center of the circle, necessary for maintaining the object on its curved trajectory.

In the case of our bird, the centripetal acceleration keeps it firm on its circular path, despite its intention to rise vertically.

• **Calculation**: Centripetal acceleration is calculated using the formula: \( a_c = \frac{v_c^2}{r} \), where \( v_c \) is the circular speed and \( r \) is the radius of the circle. For our bird example, this results in an acceleration of approximately 9.48 m/s² towards the center of its circular path.
• **Physical Meaning**: Without this inward acceleration, the bird would fly off in a straight line due to inertia. It acts like a "centripetal force" that pulls the bird towards the center, maintaining its circular path.
• **Distinction from Other Accelerations**: Unlike other forms of acceleration that might change speed, centripetal acceleration alters the direction of velocity but not its magnitude. This distinction helps in understanding why the bird's speed along its path remains constant, while its direction can continuously change.

Realizing the role of centripetal acceleration in circular motion gives us insights into how forces are balanced to maintain paths and stability in dynamic systems.

Velocity Vector Angle

When an object moves in a plane, the velocity vector describes not only the speed but also the direction of motion. The angle of this velocity vector relative to a reference line, such as the horizontal, provides insights into the overall direction of movement.

For our bird in spiral motion:

• **Components of Velocity**: The velocity vector has both horizontal (circular movement) and vertical (upward movement) components, which need to be considered to determine the overall angle.
• **Calculating the Angle**: This angle \( \theta \) can be computed using trigonometry, specifically: \( \tan \theta = \frac{v_u}{v_c} \). With the bird, the calculation gives an angle of approximately 21.8° from the horizontal.
• **Real-world Interpretation**: The angle indicates how steeply the bird is rising as it circles. A smaller angle would suggest a flatter, more lateral path, while a larger angle indicates a steeper ascent.

Understanding the concept of the velocity vector angle aids in visualizing how direction changes contribute to overall motion, and is vital in fields ranging from physics to engineering.