Question

A block of mass m1 rests on a frictionless inclined plane as shown. The inclined plane, which has mass m2, rests on a frictionless floor. Find the accelerations of both blocks with respect to the inertial reference frame provided by the floor. Hint: there are several different ways of doing this problem. You may write the equation of motion of the top block in the accelerating reference frame of the bottom block. You may instead write everything in an inertial reference frame, being careful about the relationship between vertical and horizontal positions. Alternatively, since there is no friction, energy and horizontal momentum are both conserved: you can set U = K and differentiate.

1. A block of mass m rests on a frictionless inclined plane as shown. The inclined plane, which has mass m2, rests on a frictionless floor. Find the accelerations of both blocks with respect to the inertial reference frame provided by the floor. Hint: there are several different ways of doing this problem. You may write the equation of motion of the top block in the accelerating reference frame of the bottom block. You ma,y instead write everything in an inertial reference frame, being careful about the relationship between vertical and horizontal positions. Alternatively, since there is no friction, energy and horizontal momentum are botih conserved: you can set UK and differentiate

Answer 1

To make the frame inertial we have to apply a pseudo force on the block m1.

Explanation and calculation with free body diagram for acceleration is shown below.

To make the frame inertial we have to apply a pseudo force on the block m1. Explanation and calculation with free body diagram for acceleration is shown below. Given, both the blocks are at rest initially: Now, when block of mass m_1 slides down the block of may m_2 will slide towards left and has the acceleration a_2 Free body diagram: For smaller block m_1 Forces are: (i) N = Normal force (ii) m_1 g downward (gravity) (iii) ma_2 (forward): pseudo force components of forces parallel to the incline: m_1 a_2 cos theta + m_1 g sin theta = m_1 a_1 (1) in perpendicular direction of incline N + m_1 a_2 sin theta = m_1 g cos theta (2) for block m_2 forces are: - (1) N (By the block of mass m_1) (ii) m_2 g downward N'1 (upward) by horizontal surface In horizontal direction: - N sin theta = m_2 a_2 N = m_2 a_2/sin theta (3) From eqtn (1) m_1 a_2 cos theta + m_1 g sin theta = m_1 a_1