15.13
Équation du mouvement : mouvement général du plan – Résolution de problèmes
Considérons un rouleau de pelouse d'une masse de 100 kg, d'un rayon de 0,2 mètres et d'un rayon de giration de 0,15 mètres. Une force de 200 N est app…
Considérons un rouleau à gazon de masse 100 kg, d’un rayon de 0,2 mètre et d’un rayon de giration de 0,15 mètre. Si une force de 200 N est appliquée avec un angle de 60 degrés avec l’horizontale, alors quelle est l’accélération angulaire du rouleau à gazon ?
Le coefficient de frottement statique entre le sol et le rouleau à gazon est de 0,15 et le coefficient de frottement cinétique est de 0,1.
En supposant un roulement sans glissement, le moment du point de vitesse instantanée nulle, point A, est calculé à l’aide d’une composante horizontale de la force appliquée.
Au point A, le moment d’inertie est calculé à l’aide du théorème de l’axe parallèle.
En substituant la valeur du moment d’inertie au point A dans l’équation du moment, on obtient la valeur de l’accélération angulaire.
L'hypothèse d'un roulement sans mouvement de glissement est valable si la force de frottement résultant du mouvement du centre du rouleau à gazon est inférieure à la force de frottement statique maximale.
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Q1: How do you calculate angular acceleration for a rolling object using the instantaneous center of zero velocity?
The instantaneous center of zero velocity, or point A, serves as the reference for calculating moments in rolling motion. The horizontal component of the applied force creates a moment about this point. Using the parallel axis theorem, the moment of inertia at point A is determined. Substituting this moment of inertia into the moment equation yields the angular acceleration of the rolling object.
Q2: What is the parallel axis theorem and why is it used in rolling motion problems?
The parallel axis theorem relates the moment of inertia about different axes. In rolling motion problems, it calculates the moment of inertia at the instantaneous center of zero velocity, which differs from the center of mass. This calculation is essential for applying the moment equation and determining angular acceleration when forces act on rolling objects.
Q3: When is the rolling without slipping assumption valid for a moving object?
Rolling without slipping is valid when the frictional force from the object's center motion remains lower than the maximum static frictional force. This condition ensures the object maintains contact with the surface without sliding. If the required friction exceeds the maximum static friction available, the object will slip, invalidating the rolling without slipping assumption.
Q4: How do static and kinetic friction coefficients affect rolling motion analysis?
Static friction coefficient determines the maximum frictional force available before slipping occurs, critical for validating rolling without slipping. Kinetic friction coefficient applies if slipping does occur. Both coefficients characterize the interaction between the rolling object and the surface, influencing whether the rolling without slipping assumption holds true.
Q5: What role does the radius of gyration play in calculating moment of inertia?
The radius of gyration represents the distance from the axis where the entire mass could be concentrated to produce the same moment of inertia. It simplifies moment of inertia calculations for complex shapes. Combined with mass, it allows quick determination of the moment of inertia needed for the moment equation in general plane motion problems.
Q6: How does the angle of an applied force affect the motion of a rolling object?
The angle of the applied force determines its horizontal and vertical components. Only the horizontal component contributes to the moment about the instantaneous center of zero velocity, affecting angular acceleration. The vertical component influences the normal force and thus the maximum available friction, impacting whether rolling without slipping occurs.
Q7: What steps are involved in solving a general plane motion problem with rolling constraints?
First, identify the instantaneous center of zero velocity and calculate the moment using the horizontal force component. Second, apply the parallel axis theorem to find moment of inertia at that point. Third, substitute into the moment equation to find angular acceleration. Finally, verify the rolling without slipping assumption by comparing required friction to maximum static friction.