A small block of mass m rests on the rough, sloping side of a triangular block of mass M which itself rests on a horizontal frictionless table as shown in the figure ( Eigure 1) Part A If the coefficient of static friction is ju determine the minimum horizontal force F applied to M that will cause the small block m to start moving up the incline Express your answer in terms of the given quantities and appropriate constants. AId

To determine the minimum horizontal force F required to cause the small block m to start moving up the incline, we need to consider the forces acting on the system.<br /><br />The forces acting on the small block m are:<br />1. Gravitational force (weight) acting downward, equal to mg.<br />2. Normal force acting perpendicular to the incline.<br />3. Frictional force acting parallel to the incline, opposing the motion of the block.<br /><br />The forces acting on the triangular block M are:<br />1. Gravitational force (weight) acting downward, equal to Mg.<br />2. Normal force acting perpendicular to the table.<br />3. Frictional force acting parallel to the table, opposing the motion of the block.<br /><br />To find the minimum horizontal force F required to cause the small block m to start moving up the incline, we need to consider the forces acting on the small block m.<br /><br />The frictional force acting on the small block m is given by the equation:<br />Frictional force = coefficient of static normal force<br /><br />The normal force acting on the small block m is equal to the component of the gravitational force acting perpendicular to the incline. This can be calculated using the equation:<br />Normal force = mg * cos(θ)<br /><br />where θ is the angle of the incline.<br /><br />The frictional force acting on the small block m will oppose the motion of the block and will be equal to the coefficient of static friction times the normal force.<br /><br />Therefore, the minimum horizontal force F required to cause the small block m to start moving up the incline is given by the equation:<br />F = frictional force + component of gravitational force parallel to the incline<br /><br />Substituting the values, we get:<br />F = coefficient of static friction * normal force + mg * sin(θ)<br /><br />Simplifying further, we get:<br />F = coefficient of static friction * mg * cos(θ) + mg * sin(θ)<br /><br />Therefore, the minimum horizontal force F required to cause the small block m to start moving up the incline is given by the expression:<br />F = coefficient of static friction * mg * cos(θ) + mg * sin(θ)