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Chapter 3: Problem 321

A 30 -g tire-balance weight is attached to a vertical surface of the wheel rim by means of an adhesive backing. The tire-wheel unit is then given a final test on the tire-balance machine. If the adhesive can support a maximum shear force of \(80 \mathrm{N}\), determine the maximum rotational speed \(N\) for which the weight remains fixed to the wheel. Assume very gradual speed changes.

Short Answer

Expert verified
The maximum rotational speed \( N \) depends on the radius \( r \). Calculate \( N \) using \[ N = \frac{60}{2\pi} \sqrt{\frac{80}{0.03 \cdot r}} \].

Step by step solution

01

Understand the Problem

The problem involves a 30-g weight attached to a wheel rim that is rotated at speed. We need to ensure that the shear force, due to rotation, does not exceed the adhesive's maximum capability of 80 N.
02

Identify Relations and Convert Units

The shear force exerted on the weight is due to the centripetal force, which is given by the formula \( F = m \cdot a \), where \( a \) is the centripetal acceleration \( a = r \cdot \omega^2 \). We need to express angular velocity \( \omega \) in terms of rotational speed \( N \). Also, convert mass from grams to kilograms: \( 30 \, \text{g} = 0.03 \, \text{kg} \).
03

Formulate the Equation

The centripetal force required to keep the weight moving in a circle is given by \( F = m \cdot r \cdot \omega^2 \). Substituting \( \omega = \frac{2\pi N}{60} \) (since \( N \) is in revolutions per minute, and \( \omega \) in radians per second), we have the force equation \( F = m \cdot r \cdot \left( \frac{2\pi N}{60} \right)^2 \).
04

Solve for Maximum Speed

Since the maximum shear force is 80 N, set \( F = 80 \) N in the equation: \[ 80 = 0.03 \cdot r \cdot \left( \frac{2\pi N}{60} \right)^2 \]. Solving for \( N \): \[ N = \frac{60}{2\pi} \sqrt{\frac{80}{0.03 \cdot r}} \].
05

Conclusion with Assumptions

Notice that the calculation requires the radius \( r \). Assuming you know the radius of the wheel rim where the weight is attached, you can calculate \( N \) by plugging \( r \) into the equation from Step 4.

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Key Concepts

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

Dynamics
Dynamics is a branch of mechanics focused on forces and motion. It investigates how forces impact the movement of objects. In the problem, dynamics deals with how the balance weight responds to being spun at high speeds.
  • When an object moves through a path, dynamics considers all forces acting upon it.
  • Newton's laws are fundamental in understanding dynamics. They explain how forces result in acceleration, affecting an object's speed and direction.
Understanding dynamics helps in predicting how an object will behave when influences such as shear and centripetal forces act on it.
Centripetal Force
Centripetal force keeps an object moving in a circle. It acts perpendicular to the motion and towards the center of the circle. This is critical in our tire exercise as the centripetal force must not exceed the adhesive's shear strength.
  • If this force overcomes the adhesive force, the weight will detach from the wheel.
  • Mathematically, it's calculated as: \( F = m \cdot a \), where \( a = r \cdot \omega^2 \).
It is essential for any moving object in circular motion but could lead to failure if not properly managed.
Shear Force
Shear force is a force that acts parallel to a surface. In engineering mechanics, this force can cause deformation. The adhesive on the wheel rim experiences shear force, trying to slide the weight off when the wheel spins.
  • The maximum shear force the adhesive can withstand is 80 N.
  • Determining if this is sufficient involves ensuring the centripetal force remains below this threshold.
This ensures the balance weight stays secure and emphasizes the importance of using adhesive capable of handling these forces.
Rotational Speed
Rotational speed, denoted by \( N \), is how fast an object rotates. It's typically measured in revolutions per minute (rpm). In our scenario, rotational speed affects the centrifugal force driving the shear on the tire weight.
  • Higher speeds increase the force acting on the weight.
  • A crucial part of the problem is finding the maximum speed the weight can endure without detaching.
  • It involves converting revolutions to radians for accurate calculations.
Understanding rotational speed helps in predicting the performance and stability of rotating objects, ensuring safety and functionality.

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Most popular questions from this chapter

The 0.1 -lb projectile \(A\) is subjected to a drag force of magnitude \(k v^{2},\) where the constant \(k=\) \(0.0002 \mathrm{lb}-\mathrm{sec}^{2} / \mathrm{ft}^{2} .\) This drag force always opposes the velocity \(\mathbf{v} .\) At the instant depicted, \(v=100 \mathrm{ft} / \mathrm{sec}\) \(\theta=45^{\circ},\) and \(r=400 \mathrm{ft} .\) Determine the corresponding values of \(\ddot{r}\) and \(\ddot{\theta}\).

The slider of mass \(m_{1}=0.4 \mathrm{kg}\) moves along the smooth support surface with velocity \(v_{1}=5 \mathrm{m} / \mathrm{s}\) when in the position shown. After negotiating the curved portion, it moves onto the inclined face of an initially stationary block of mass \(m_{2}=2 \mathrm{kg}\) The coefficient of kinetic friction between the slider and the block is \(\mu_{k}=0.30 .\) Determine the velocity \(v^{\prime}\) of the system after the slider has come to rest relative to the block. Neglect friction at the small wheels, and neglect any effects associated with the transition.

The third stage of a rocket fired vertically up over the north pole coasts to a maximum altitude of \(500 \mathrm{km}\) following burnout of its rocket motor. Calculate the downward velocity \(v\) of the rocket when it has fallen \(100 \mathrm{km}\) from its position of maximum altitude. (Use the mean value of \(9.825 \mathrm{m} / \mathrm{s}^{2}\) for \(g\) and \(6371 \mathrm{km}\) for the mean radius of the earth.)

A projectile is launched from the north pole with an initial vertical velocity \(v_{0} .\) What value of \(v_{0}\) will result in a maximum altitude of \(R / 3 ?\) Neglect aerodynamic drag and use \(g=9.825 \mathrm{m} / \mathrm{s}^{2}\) as the surface-level acceleration due to gravity.

The slotted arm revolves in the horizontal plane about the fixed vertical axis through point \(O .\) The 3-lb slider \(C\) is drawn toward \(O\) at the constant rate of 2 in./sec by pulling the cord \(S\). At the instant for which \(r=9\) in., the arm has a counterclockwise angular velocity \(\omega=6\) rad/sec and is slowing down at the rate of 2 rad \(/\) sec \(^{2}\). For this instant, determine the tension \(T\) in the cord and the magnitude \(N\) of the force exerted on the slider by the sides of the smooth radial slot. Indicate which side, \(A\) or \(B\), of the slot contacts the slider.
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