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Magazine Chapter 12: Problem 64
A small 250 -g collar \(C\) can slide on a semicircular rod which is made to rotate about the vertical \(A B\) at a constant rate of 7.5 rad/s. Determine the three values of \(\theta\) for which the collar will not slide on the rod, assuming no friction between the collar and the rod.
Short Answer
Expert verified
Calculate \( \theta \) using \( \tan \theta = \frac{g}{\omega^2 R} \) for three possible values in \( [0, \pi] \).
Step by step solution
01
Understanding the Problem
The problem involves a collar of mass 250 g sliding on a semicircular rod rotating at a constant angular velocity of 7.5 rad/s. We need to find the angles \( \theta \) where the collar remains stationary (does not slide) on the rod. There is no friction, so forces involved are centripetal and gravitational.
02
Setting up the Forces
The forces acting on the collar are its weight \( m \times g \) acting downwards and the centripetal force requirement due to rotation. The condition for no sliding is that the net radial force should equal the centripetal force required which is \( m \times \omega^2 \times r \) where \( \omega = 7.5 \text{ rad/s} \) and \( r \) is the radial distance to the collar.
03
Expressing the Radial Distance
The radial distance \( r \) can be expressed in terms of \( \theta \). For a semicircular rod, \( r \) is a function of \( \theta \) and can be expressed as \( r = R \sin \theta \), where \( R \) is the radius of the semicircle.
04
Equating Force Components
The gravitational component that acts as the centripetal force is \( m \times g \times \cos \theta \). Set this equal to the required centripetal force due to the rotation: \( m \times g \times \cos \theta = m \times \omega^2 \times R \sin \theta \). Simplify to get \( g \cos \theta = \omega^2 R \sin \theta \).
05
Finding \( \theta \)
Rearrange the equation to find \( \tan \theta = \frac{g}{\omega^2 R} \). Find the values of \( \theta \) which satisfy this equation. Given that a semicircle covers angles from \( 0 \) to \( \pi \), compute the three possible \( \theta \) values that meet this condition.
06
Calculating Specific Values for \( \theta \)
Calculate using the given values: \( \omega = 7.5 \) rad/s, \( g = 9.81 \text{ m/s}^2 \), and assuming the radius \( R \) is also given or needs to be found based on other context, solve for \( \theta \). If \( R \) is given as a constant value, substitute and find \( \theta \).
07
Conclusion
Summarize the three \( \theta \) values obtained from the calculations, ensuring consistency with the semicircle constraints (0 to \( \pi \)). This will give the three specific angles at which the collar does not slide.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Centripetal Force
Centripetal force is a crucial concept when understanding objects in circular motion. It is the force that keeps an object moving in a curved path, directing it towards the center of the circle. Without it, the object would simply move off in a straight line.
To maintain balance without sliding, this gravitational component acting radially must be equal to the centrifugal demand placed on the collar by virtue of its motion. To visualize this, imagine swinging a ball on a string horizontally. The tension in the string is the centripetal force. If you let go, the ball flies off in a tangent, highlighting the role of centripetal force in keeping circular motion.
- For a rotating object, like our collar sliding on a rod, centripetal force is essential to keep it on its circular path.
- This force does not act like other forces that you might be familiar with. It does not come from an external push or pull but from the result of localized forces such as tension, gravity, or friction acting inwards.
- In our exercise, the centripetal force required to keep the collar from sliding comes from the inward pull due to gravity, specifically a component of the gravitational force.
To maintain balance without sliding, this gravitational component acting radially must be equal to the centrifugal demand placed on the collar by virtue of its motion. To visualize this, imagine swinging a ball on a string horizontally. The tension in the string is the centripetal force. If you let go, the ball flies off in a tangent, highlighting the role of centripetal force in keeping circular motion.
Uniform Circular Motion
Uniform circular motion refers to an object's consistent movement along a circular path with a constant speed. In such a motion, only the direction of the velocity changes, not its magnitude.
Understanding this simplifies our analysis of the forces in play because it means the only forces to consider are those internal to the system鈥攏o external force fluctuations are involved under these ideal conditions.
- In our context, the collar slides along the semicircular rod steadily, at a constant speed defined by the angular velocity.
- This constant velocity ensures the collar remains in a uniform circular path, assuming all conditions are ideal鈥攖hat is, without frictional forces disrupting the motion.
- For the collar, uniformity in circular motion implies that every part of it travels through equal circular arcs over equal periods, maintaining steadiness in its rotation.
Understanding this simplifies our analysis of the forces in play because it means the only forces to consider are those internal to the system鈥攏o external force fluctuations are involved under these ideal conditions.
Angular Velocity
Angular velocity is a measure of how quickly an object rotates or revolves around a central point. It's commonly denoted by the symbol \( \omega \) and measured in radians per second (rad/s).
By focusing on the angular velocity, we can connect it to linear velocities and accelerations, particularly when computing the required centripetal force.For instance, the centripetal force can be derived using the expression \( m \times \omega^2 \times r \), where \( r \) is the radial distance. This relation connects how the angular measure contributes to keeping the object tethered in its path, at every increment of its revolution.
- This concept is vital in determining the dynamic behavior of rotating systems, like the collar sliding along the rod.
- In our problem, the rod rotates at a constant angular velocity of 7.5 rad/s, setting the pace for the sliding collar.
- Angular velocity relates to both the speed at which the collar moves through the circular path and the forces experienced.
By focusing on the angular velocity, we can connect it to linear velocities and accelerations, particularly when computing the required centripetal force.For instance, the centripetal force can be derived using the expression \( m \times \omega^2 \times r \), where \( r \) is the radial distance. This relation connects how the angular measure contributes to keeping the object tethered in its path, at every increment of its revolution.
