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Chapter 3: Problem 70

The slotted arm \(O A\) rotates about a fixed axis through \(O\). At the instant under consideration, \(\theta=\) \(30^{\circ}, \dot{\theta}=45 \operatorname{deg} / \mathrm{s},\) and \(\ddot{\theta}=20 \mathrm{deg} / \mathrm{s}^{2} .\) Determine the forces applied by both arm \(O A\) and the sides of the slot to the 0.2 -kg slider \(B\). Neglect all friction, and let \(L=0.6 \mathrm{m} .\) The motion occurs in a vertical plane.

Short Answer

Expert verified
The horizontal force on slider B from the slot is \(F_h = \frac{0.12\pi^2}{16} \cos(\frac{\pi}{6})\) N, and the vertical force by arm OA is \(F_v = \frac{0.12\pi}{9} + \frac{0.12\pi^2}{16} \sin(\frac{\pi}{6}) - 1.962\) N.

Step by step solution

01

Understanding the Problem

In this exercise, a slotted arm OA rotates around a fixed axis, causing a 0.2 kg slider, B, to move. We need to determine the forces exerted on B by the arm OA and the slot sides at a given rotational angle, angular velocity, and angular acceleration. Given values are: \(\theta = 30^{\circ}, \dot{\theta} = 45^{\circ}/s, \ddot{\theta} = 20^{\circ}/s^{2}\), and the arm length \(L = 0.6\, m\).
02

Convert Angular Quantities to Radians

Angular motion is usually measured in radians for calculations. Convert the given angles:\[\theta = 30^{\circ} = \frac{\pi}{6}\, radians\]\[\dot{\theta} = 45^{\circ}/s = \frac{45\pi}{180} = \frac{\pi}{4}\, rad/s\]\[\ddot{\theta} = 20^{\circ}/s^{2} = \frac{20\pi}{180} = \frac{\pi}{9}\, rad/s^{2}\]
03

Calculate Tangential Acceleration

The tangential acceleration \(a_t\) can be found by multiplying angular acceleration with length:\[a_t = L \cdot \ddot{\theta} = 0.6 \cdot \frac{\pi}{9} = \frac{0.6\pi}{9}\, m/s^{2}\]
04

Calculate Radial (Centripetal) Acceleration

The radial acceleration \(a_r\) is calculated using the formula:\[a_r = L \cdot \dot{\theta}^2 = 0.6 \cdot \left( \frac{\pi}{4} \right)^2 = \frac{0.6\pi^2}{16}\, m/s^2\]
05

Determine Forces in Horizontal and Vertical Directions

Forces are determined using the mass of the slider (0.2 kg). The tangential force is:\[F_t = m \times a_t = 0.2 \times \frac{0.6\pi}{9} = \frac{0.12\pi}{9}\, N\] The radial force is:\[F_r = m \times a_r = 0.2 \times \frac{0.6\pi^2}{16} = \frac{0.12\pi^2}{16}\, N\]
06

Calculate Components of Resultant Force

Using trigonometry, resolve the radial and tangential forces. The forces exerted by the arm \(OA\) and the slot on the slider have horizontal and vertical components:- Horizontal component (centripetal force): - Horizontal force by the slot sides: \(F_h = F_r \cdot \cos(\theta)\) \(F_h = \frac{0.12\pi^2}{16} \times \cos(\frac{\pi}{6})\)- Vertical component (gravitationally opposed net force): - Vertical force by the arm: \(F_v = F_t + F_r \cdot \sin(\theta) - mg\) \(F_v = \frac{0.12\pi}{9} + \frac{0.12\pi^2}{16} \times \sin(\frac{\pi}{6}) - 0.2 \times 9.81\)
07

Solution

The forces on B are the components calculated: 1. Force applied by the sides of the slot horizontally. 2. Force applied by the arm vertically resolved due to its motion and gravitational influence.

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Key Concepts

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

Angular Motion
Angular motion refers to the movement of a body about a specific point or axis in a rotational path. In this scenario, the slotted arm OA rotates around the fixed point O, causing the slider B to move in a slotted path. The concepts related to angular motion include:
  • Angular displacement: the angle through which a point or line has been rotated in a specified sense about a specified axis.
  • Angular velocity: the rate of change of angular displacement, measured in degrees per second or radians per second.
  • Angular acceleration: the rate of change of angular velocity, typically measured in degrees per second squared or radians per second squared.
Understanding these principles helps us analyze how moving parts, like the arm OA in this problem, affect connected objects, such as the slider B.
Tangential Acceleration
Tangential acceleration is a measure of how quickly the velocity of an object moving along a circular path changes over time. It is calculated using the angular acceleration and the radius of the circular path.
  • The formula for tangential acceleration is: \( a_t = L \cdot \ddot{\theta} \), where \( L \) is the length from the axis of rotation to the point of interest, and \( \ddot{\theta} \) is the angular acceleration.
  • In our example, with \( L = 0.6 \) m and \( \ddot{\theta} = \frac{\pi}{9} \) rad/s虏, the tangential acceleration is \( \frac{0.6\pi}{9} \) m/s虏.
This concept is crucial because it directly influences the tangential force exerted by a moving arm on the slider, affecting how the slider moves along the slot.
Centripetal Force
Centripetal force is the inward force required for an object to follow a circular path. It arises due to centripetal acceleration which points towards the center of rotation.
  • The formula for radial or centripetal acceleration is: \( a_r = L \cdot \dot{\theta}^2 \), where \( \dot{\theta} \) is the angular velocity.
  • In this problem, \( L = 0.6 \) m and \( \dot{\theta} = \frac{\pi}{4} \) rad/s, leading to \( a_r = \frac{0.6\pi^2}{16} \) m/s虏.
Centripetal force ensures that the slider stays in its path along the slot as the arm rotates. It's critical to account for this when calculating the forces experienced by the slider.
Rigid Body Kinematics
Rigid body kinematics involves the study of motion without considering forces or torques specifically. For a rigid body such as the slotted arm OA, this involves:
  • Translational motion: All points of the body move in parallel along a straight or curved path.
  • Rotational motion: Movement around a single axis where all particles of the body have the same angular velocity and angular acceleration.
In this exercise, understanding the rigid body kinematics allows us to observe how the rotation of the arm OA influences the motion of the slider B. It is important to calculate positions, velocities, and accelerations effectively during analysis.
Rotational Motion
Rotational motion is the rotation of a body around a central axis. This system includes angular displacement, velocity, and acceleration.
  • In our problem, the slotted arm's rotation about its fixed axis causes the slider to move within the slot.
  • Rotational motion is fundamental in predicting how arm OA will transfer motion and forces to slider B.
Mastering the basics of rotational motion is vital for anyone dealing with systems involving rotating elements. By understanding these principles, we can effectively calculate the resulting forces and accelerations impacting various parts of the system.

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Most popular questions from this chapter

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