A uniform rod with a weight of 10 lb is pin-supported at point A from a roller that rides on a horizontal track. If the rod is initially at rest and a horizontal force of [tex]$F = 15 \text{ lb}$[/tex] is applied to the roller, determine the acceleration of the roller. Neglect the mass of the roller and its size [tex]d[/tex] in the computations.

(Answer: [tex]a_A = 193 \text{ ft/s}^2[/tex])

The acceleration of the roller can be calculated using Newton's Second Law, but the information provided is insufficient to perform the calculation accurately. The assumption that there is no rotation—implied by ignoring the roller's mass and size—leads to a straightforward application of F=ma, but the dynamics of the rod's rotation and the pin support are not addressed, which would impact the actual acceleration.

Explanation:

The question is asking to determine the acceleration of a roller when a horizontal force is applied to a uniform rod that is supported by a pin at one end. To find the acceleration of the roller, we must use Newton's second law, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration (F = ma). The neglect of the mass and size of the roller implies that all of the force acts to accelerate the rod horizontally, meaning we should consider the rod's entire weight as part of the resistance to the applied force.

However, without information on the length of the rod and its moment of inertia, an assumption is needed that the force does not cause rotation, which we can presume is the intent given that we are disregarding the size of the roller. Given these constraints, the acceleration would simply be the force divided by the weight converted to mass (using g = 32.2 ft/s2 to convert pounds to slugs for mass). However, the answer provided suggests that the dynamics of the rod, including its rotation and the way force is distributed due to the pin support, play a role in determining the correct acceleration.

Since the specific mechanism of how the force contributes to linear acceleration versus rotational acceleration is not described, and the question requires information not provided (such as distribution of mass and dimensions), a direct calculation of the acceleration cannot be concluded accurately here. The provided answer of aA = 193 ft/s2 likely comes from a more detailed calculation incorporating these additional dynamics. Typically, one would need to apply the principles of rotational dynamics and the equations governing the motion of a rigid body with a fixed axis to arrive at the correct acceleration.