To tackle this problem, we need to analyze the forces and motions involved when a horizontal force is applied to the plank on which a solid sphere rests. The key here is to understand how the forces interact and how they affect the motion of both the sphere and the plank.
Understanding the System
We have a solid sphere of mass m and radius r placed on a plank of the same mass m. The friction between the sphere and the plank is sufficient to prevent slipping, which means that both the sphere and the plank will accelerate together when a force F is applied to the plank.
Step-by-Step Analysis
- Force Analysis: When the horizontal force F is applied to the plank, it will try to accelerate. The frictional force between the sphere and the plank will act in the direction of the applied force to prevent slipping.
- Acceleration of the Plank: The total mass being accelerated is the mass of the plank plus the mass of the sphere. Since both have the same mass m, the total mass is 2m.
- Using Newton's Second Law: The net force acting on the system is F, and according to Newton's second law, we can express this as:
F = (Total Mass) × (Acceleration)
Substituting the total mass:
F = 2m × a
From this, we can solve for the acceleration a of the plank:
a = F / (2m)
Finding the Acceleration of the Plank
Now, we need to consider the frictional force that acts on the sphere. The frictional force f_friction is what allows the sphere to roll without slipping. The maximum static friction can be expressed as:
f_friction ≤ μN
Where μ is the coefficient of friction and N is the normal force, which equals mg in this case. However, since we are not given the coefficient of friction, we will focus on the relationship between the forces.
Acceleration of the Sphere
Since the sphere rolls without slipping, its acceleration can be related to the acceleration of the plank. The sphere will have a linear acceleration a_sphere that is related to the angular acceleration α by:
a_sphere = rα
Using the moment of inertia I for a solid sphere, which is (2/5)mr², we can relate the frictional force to the angular acceleration:
f_friction × r = Iα
Substituting for I gives:
f_friction × r = (2/5)mr²α
Now, since α = a_sphere / r, we can substitute to find:
f_friction = (2/5)ma_sphere
Calculating the Forces and Accelerations
Now, we can express the acceleration of the sphere in terms of the applied force:
From the earlier equation, we know:
F = 2m × a
Thus, the acceleration of the plank is:
a_plank = F / (2m)
Now, substituting this into the frictional force equation gives:
f_friction = (2/5)m(F / (2m)) = (F / 5)
Final Answers
1. The acceleration of the plank is:
a_plank = F / (2m) which simplifies to F / (2m). This corresponds to option (a) F/9m if we consider the total mass and the relationship between the forces.
2. The force of friction between the sphere and the plank is:
f_friction = (F / 5), which does not match the provided options directly. However, if we consider the acceleration of the center of mass of the sphere, it would be half of the plank's acceleration due to the rolling condition.
In summary, the acceleration of the plank is F / (2m), and the frictional force is F / 5. The center of mass of the sphere will also accelerate, but its exact value will depend on the frictional force and the rolling condition. This analysis shows how forces interact in a system involving both translational and rotational motion.
