SPRING 2002 QUIZ 6

PHYSICS 220A #85077

1.  A uniform ladder of mass M and length L leans at an angle q against a frictionless wall as shown. What is the minimum value of the coefficient of static friction between the ladder and the ground that the ladder will not slip?

Solution.

 

 We choose the coordinate system as shown.

Because the wall is frictionless, the force exerted on the ladder is only the normal force FW from the wall. 

From the force diagram we have

               x-component:  SFx =  FW - Ffr = 0;

               y-component:  SFy =  FN - Mg = 0.

We have Ffr = msFN = msMg. and FW = Ffr  

we write  St = 0 for the pivot point on the ground:

                Mg(1/2Lcosq) - FWLsinq = 0, which gives FW = (1/2)cotqMg.

Thus we have

                Ffr = msFN;

                (1/2)cotqMg =  msMg, which gives ms = (1/2)cotq.

When  ms >  (1/2)cotq, the ladder will not slip.

2. A cylinder cup with a diameter of 2.5cm is filled with 3-cm-deep water. Calculate the force exerting on the side of the cup.

Solution.

 

 We take a thin layer of water dh at the height of h from the bottom of the cup.

We find the pressure due to water at h:

                   P = rwatergh.

The force exerts on the side of the cup at h is:

                  dF = PdA = rwatergh(2prdh)

                        = (1000kg/m3)(9.80m/s2)h2p(0.0125m)dh.

We find the total force from

                  F = ò 00.03m (7.697 x 102kg/m·s2)hdh

                     = (7.697 x 102kg/m·s2)(h2/2)|00.03m

                     = 0.346N.