91Ó°ÊÓ

A Ride Inside a Tractor Tire. You and your friends plan to roll down a hill on the inside of 600 -pound tractor tire (diameter \(D=1.80 \mathrm{m}\) ). The hill is inclined at an angle of \(25.0^{\circ}\) and you initially plan to start from a distance \(L=100 \mathrm{m}\) up the hill, but decide to first check whether it will be safe. (a) Assuming the masses of the tire and your 105 -pound body are concentrated at the outer rim of a thin-walled cylinder/hoop, what is the effective acceleration your body experiences at the bottom of the hill where your angular speed is greatest, i.e., how many "g's" will you experience? Assuming the human body can withstand a g-force of \(8.00 \mathrm{g}\) 's \(\left(1 \mathrm{g}=9.80 \mathrm{m} / \mathrm{s}^{2}\right),\) is it safe to make the ride from \(100 \mathrm{m}\) up the hill? (b) What is the maximum starting distance \(\left(L_{\text {max }}\right)\) up the hill that is safe?

Step by step solution

01

Convert Mass to Kilograms

The weight of the tire is 600 pounds, and your weight is 105 pounds. First, convert these weights to kilograms:1 pound = 0.453592 kg.So, the mass of the tire is \(600 \times 0.453592 \approx 272 \text{ kg}\) and your mass is \(105 \times 0.453592 \approx 47.63 \text{ kg}\).

02

Calculate Total Mass and Moment of Inertia

Since both the masses are concentrated at the outer rim, the total mass \(M\) is the sum of your mass and the tire's mass:\[ M = 272 + 47.63 = 319.63 \text{ kg}\]The moment of inertia \(I\) of a hoop is \(I = MR^2\), where \(R\) is the radius:With the diameter given as 1.80 m, the radius \(R = \frac{1.80}{2} = 0.90 \text{ m}\).

03

Use Energy Conservation to Find Final Velocity

Use the conservation of mechanical energy to find the velocity at the bottom:Potential energy at the top = Kinetic energy at the bottom. \[ MgL\sin(\theta) = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2 \]Since \(\omega = \frac{v}{R}\), substitute \(I = MR^2\) to get:\[ MgL\sin(\theta) = Mv^2 \]Solve for \(v\):\[ v = \sqrt{2gL\sin(\theta)} \]

04

Calculate Acceleration Due to Rotation

Upon reaching the bottom, the centripetal acceleration is calculated by:\[ a = \frac{v^2}{R} \]

05

Determine G-force Experienced

Convert this into g-force:\[ \text{g-force} = \frac{a}{g} = \frac{\frac{2gL\sin(\theta)}{R}}{g} = \frac{2L\sin(\theta)}{R} \]

06

Check Safety for 100 m Start

Plug in the values to check safety:\(L = 100 \text{ m}, \theta = 25^\circ, R = 0.90 \text{ m}\).\[ \text{g-force} = \frac{2 \times 100 \times \sin(25^\circ)}{0.90} \approx 78.52 \text{ g's} \]Since \(78.52 \text{ g's} > 8.00 \text{ g's} \), the ride is not safe.

07

Calculate Maximum Safe Distance

To find the maximum starting distance \(L_{\text{max}}\) that is safe, limit the g-force to 8.0 g's:\[ 8.0 = \frac{2L_{\text{max}}\sin(25^\circ)}{R} \]Solve for \(L_{\text{max}}\):\[ L_{\text{max}} = \frac{8.0 \times R}{2\sin(25^\circ)} \approx 11.47 \text{ m} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics

Kinematics is the branch of physics that studies the motion of objects without considering the causes of motion. In this exercise, understanding kinematics helps us describe the motion of the tire as it rolls down the hill. Key components include:

• Distance and Displacement: Distance is the total path length traveled, while displacement is the straight line distance from the initial to the final position.
• Velocity and Speed: Velocity is a vector quantity (it has both magnitude and direction), whereas speed is scalar (only magnitude). The exercise uses potential to kinetic energy conversion to calculate the velocity at the hill's bottom.
• Acceleration: Acceleration occurs when there is a change in velocity. It can be constant (as seen in free fall) and is crucial in connecting kinematic equations to calculate the body's acceleration inside the tire.

Understanding these core aspects allows us to predict how fast the tire accelerates and what velocity it reaches at the bottom. As shown, using the formula \[ v = \sqrt{2gL\sin(\theta)} \]we determined velocity, crucial for further calculations like finding the centripetal acceleration.

Energy Conservation

Energy conservation is a fundamental concept in physics that states energy in a closed system remains constant. When the tire rolls down the hill, gravitational potential energy is converted into kinetic energy.The key steps involve:

• Potential Energy: Initially at the starting point, the potential energy is given by \(MgL\sin(\theta)\), where \(M\) is the mass, \(g\) is gravity, and \(\theta\) is the slope angle.
• Kinetic Energy: As the tire accelerates down the hill, potential energy is transformed into kinetic energy, both translational \((\frac{1}{2}Mv^2)\) and rotational \((\frac{1}{2}I\omega^2)\).
• Energy Conservation Equation: This principle results in the energy balance equation \[ MgL\sin(\theta) = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2 \]used to derive the velocity formula. This illustrates the transformations between energy types without loss.

The exercise teaches how energy conservation allows us to predict motion end-states by comparing initial energy to final energy.

G-Force Calculation

G-force is a measure of acceleration felt as weight. It's important in determining if the human body can safely endure the forces during motion inside a tire.Key concepts include:

• What is G-force? G-force describes the acceleration resulting from force that defies gravity, represented in multiples of \(g\) where \(g = 9.80 \text{ m/s}^2\).
• Acceleration Calculation: Once the kinematic and energy conservation equations provide speed, centripetal acceleration is derived using \[ a = \frac{v^2}{R} \]which affects the g-force felt.
• Safety Considerations: By converting acceleration into g-force \[ \text{g-force} = \frac{a}{g} \], we can compare these against safe human g-force limits e.g., 8.00 g's in this exercise.

Through calculations, the exercise determines the g-force exceeds safe thresholds for the hill's full distance, making it imperative to recalculate maximum safe starting points. This illustrates evaluating physical limitations and safety in everyday physics applications.