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Figure 4: Top view of force table. (3) Figure 5 is an inclined-plane system that will be stud- ied in the first part of this experiment. As labelled in the figure, the x (y) direction is parallel (perpendicular) to the inclined plane, and the gravitational acceleration is downward. If the hanging mass m is too small, the block mass M on the inclined plane slides down. When the hanging mass is gradually increased to a lower-bound value mi such that the block mass M just stops sliding down, the forces on M satisfy the equilibrium conditions: )a Figure 5: Inclined-plane system.mhg + f-Mg sin α = 0, N_Mg cos α = 0, in x direction, ¡n y direction, Here f is a friction force and N is a normal force. Both of them are exerted by the inclined plane. If we keep increasing the hanging mass to a higher-bound value m2 such that the block mass M is just about to slide up (but not moving yet), the forces on M satisfy the equi- librium conditions (here we assume that the magnitudes of f and N remain the same): m28-f-Mg sin α 0, in x direction, N -Mg cosa 0, in y direction, i. Use 4 vectors Mg, m1g, N, and f to draw the free-body diagram for mass M in the case corresponding to Eq. (4). How does mass M move if there is no friction (f 0)? ii. Use 4 vectors Mg, m2g, N, and f to draw the free-body diagram in the case corre- sponding to Eq. (5). How does mass M move if there is no friction (f 0)? ii. Combing the x-direction conditions in Eqs. (4) and (5), please show that m1 + m2 sin α--2M m2 g. We will experimentally examine Eq. (6) in the lab. Note that to verify Eq. (6) we do not need to know the details about gravitational acceleration g or friction f of the system.

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