Problem 47 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A jet pilot takes his aircraft in a vertical loop (Fig. \(5-43)\). ( \(a\) ) If the jet is moving at a speed of \(1200 \mathrm{~km} / \mathrm{h}\) at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed \(6.0 \mathrm{~g}\) 's. \((b)\) Calculate the \(78-\mathrm{kg}\) pilot's effective weight (the force with which the seat pushes up on him) at the bottom of the circle, and \((c)\) at the top of the circle (assume the same speed).

Step-by-Step Solution

Answer

(a) Minimum radius ≈ 1890.43 m; (b) Effective weight at bottom ≈ 1223.76 N; (c) Effective weight at top ≈ -305.04 N.

Step 1: Convert Speed from km/h to m/s

The speed of the jet is given as \(1200 \text{ km/h}\). We convert it to meters per second by using the conversion factor \(1 \text{ km/h} = \frac{5}{18} \text{ m/s}\). \[\text{Speed (in m/s)} = 1200 \times \frac{5}{18} = 333.33 \text{ m/s}\]

Step 2: Calculate Minimum Radius for Centripetal Acceleration

We use the formula for centripetal acceleration \(a_c = \frac{v^2}{r}\), where \(a_c\) is the centripetal acceleration, \(v\) is the velocity, and \(r\) is the radius of the circle. We're given that \(a_c\) should not exceed \(6g\), where \(g = 9.8 \text{ m/s}^2\).\[6g = 6 \times 9.8 = 58.8 \text{ m/s}^2\]To find the minimum radius:\[\frac{v^2}{r} \leq 58.8\quad\Rightarrow\quad r \geq \frac{v^2}{58.8}\quad\Rightarrow\quad r \geq \frac{(333.33)^2}{58.8} \approx 1890.43 \text{ m}\]

Step 3: Calculate Effective Weight at the Bottom of the Circle

At the lowest point, the centripetal force is provided by the normal force \(N_b\) (effective weight) minus the gravitational force \(mg\).\[N_b - mg = m\frac{v^2}{r}\]\[N_b = mg + m\frac{v^2}{r}\]Substitute the values:\[N_b = 78 \times 9.8 + 78 \times \frac{(333.33)^2}{1890.43}\quad\Rightarrow\quad N_b = 764.4 + 459.36 = 1223.76 \text{ N}\]

Step 4: Calculate Effective Weight at the Top of the Circle

At the top of the circle, the centripetal force is provided by the gravitational force \(mg\) plus the normal force \(N_t\). \[N_t + mg = m\frac{v^2}{r}\]\[N_t = m\frac{v^2}{r} - mg\]Substitute the values:\[N_t = 78 \times \frac{(333.33)^2}{1890.43} - 78 \times 9.8\quad\Rightarrow\quad N_t = 459.36 - 764.4 = -305.04 \text{ N}\]Negative value indicates the pilot's effective weight is directed downwards as he is supported primarily by his harness or restraints.

Key Concepts

Vertical Loop Motion

Vertical loop motion involves an object moving through a circular path that is oriented vertically. Imagine a jet zooming through a gigantic vertical circle.
The motion is fascinating and is a fantastic example of physics in action. While moving along a loop, the jet and its pilot experience forces due to gravity and their motion.
In particular, the jet experiences centripetal force, which keeps it moving along the curve of the loop. This force is constantly pulling toward the center of the circle, changing the direction of the jet's velocity.

When the jet is at different points in the loop, the forces acting on the pilot vary. At the bottom of the loop, the pilot feels heavier due to the additional centripetal force necessary to keep moving in a circle
(the force is directed upwards).
At the top of the loop, the gravitational force and centripetal force work together, and the pilot might feel lighter.

Effective Weight

Effective weight is a critical concept in understanding forces in motion, like in vertical loops. It's the apparent weight an object feels due to the combined effects of gravitational force and motion.
This can be different from the actual gravitational force acting on the object.

At the lowest point in a vertical loop, the effective weight is influenced by the centripetal force needed to maintain circular motion. Here, the pilot feels an effective weight that's significantly more than his regular weight because of the upward force acting on him,
which is greater than the downward gravitational force.
Contrasting this, at the highest point in the loop, the pilot's effective weight might even be a reduced force or negative if the centripetal force isn't counteracted completely by gravity.

• At the bottom: Effective weight is higher.
• At the top: Effective weight decreases, possibly feeling negative.

Circular Motion

Circular motion is when an object moves in a circular path at a constant speed. It's characterized by a central force that keeps the object in its curved path.
This central force is known as centripetal force, and without it, the object would fly off in a straight line due to inertia.

In the jet pilot's scenario, the aircraft moves through a vertical circular path where the centripetal force plays a vital role. It's calculated by the formula for centripetal acceleration:

• Formula: \( a_c = \frac{v^2}{r} \)
• \( v \) is the speed of the object moving,
• \( r \) is the radius of the circular path.

Ensuring the centripetal acceleration does not exceed a certain value, like 6g's for our jet pilot, ensures safety and physical integrity.
This involves calculating the minimum radius for the loop when the speed is constant.
Circular motion isn't just a fascinating concept; it's essential for real-world applications like jet maneuvers.

Physics Problem Solving

Physics problem solving is all about breaking apart the complexities of a problem into understandable pieces.
By understanding principles like forces, motions, and transitions, we can solve intricate problems systematically.

When tackling problems involving vertical loop motion, like the one with the jet pilot, it's essential to:

• Convert units carefully for precision (like speed conversion from km/h to m/s).
• Use appropriate formulas for centripetal force and acceleration.
• Consider all forces acting on the object, such as gravity and normal force.

Using these strategic steps, we can determine quantities like effective weight at distinct loop positions.
It helps predict real-world scenarios accurately. Additionally, mastering physics problem solving enhances logical thinking and application of theoretical concepts to practical situations.