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Chapter 10: Problem 67

Each wheel of a \(320-\mathrm{kg}\) motorcycle is \(52 \mathrm{cm}\) in diameter and has rotational inertia \(2.1 \mathrm{kg} \cdot \mathrm{m}^{2} .\) The cycle and its \(75-\mathrm{kg}\) rider are coasting at \(85 \mathrm{km} / \mathrm{h}\) on a flat road when they encounter a hill. If the cycle rolls up the hill with no applied power and no significant internal friction, what vertical height will it reach?

Short Answer

Expert verified
The vertical height the motorcycle will reach can be calculated using the conservation of energy concept, converting the initial translational and rotational kinetic energies into gravitational potential energy. The specific height can be determined by following the outlined steps and solving the final equation.

Step by step solution

01

Convert Velocity from km/h to m/s.

The velocity given is 85km/h, it needs to be converted to m/s for the calculation since international standard units should be used. Use the conversion factor \( \frac{1000m}{1km} \times \frac{1h}{3600s} \). Multiplying \(85 \times \frac{1000}{3600}\) gives roughly \(23.6 m/s\).
02

Calculate translational and rotational kinetic energies.

The translational kinetic energy can be given by the formula \(KE_{trans}=\frac{1}{2}m_{total}v^2\) where \(m_{total} = (320 + 75) = 395 kg\) is the total mass and \(v = 23.6 m/s\) is the velocity. The rotational kinetic energy for each wheel can be calculated by \(KE_{rot} = \frac{1}{2}I \omega^2\) where \(I = 2.1 kg.m^2\) is the moment inertia and \(\omega\) is angular velocity. Since we have the linear velocity (\(v = \omega \times r\)), \(\omega = \frac{v}{r}\). Substituting the values, we get the \(KE_{rot}\) for each wheel. We multiply it by 2 as there are two wheels.
03

Total initial kinetic energy.

Sum the translational kinetic energy and the two rotational kinetic energies to get the total initial kinetic energy.
04

Conservation of energy to calculate height.

Since we do not have any other external energy or significant internal friction, the total kinetic energy will be converted into gravitational potential energy when the cycle moves up the hill which can be given by \(PE = m_{total} \times g \times h\), where \(g = 9.8 m/s^2\) is gravitational acceleration and \(h\) is height reached by the motorcycle. Given the total potential energy equals the total kinetic energy, we can solve the equation for \(h\).

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