Physics 111 HW 14

Due Friday, 26 June 2015

S01. A diving board 3.00 m long is supported at a point 1.00 m from the end, and a diver weighing 500 N stands at the free end (see figure). The diving board is of uniform cross section and weighs 280 N.

a) Find the force at the support point (1 m from the left end of the board).

b) Find the force at the end of the board that is held down.

S03. The horizontal beam in the figure weighs 150 N and its mass distribution is uniform.

a) Find the tension in the cable.

b) Find the horizontal and vertical components of the force the hinge exerts on the board.

S04. A non-uniform fire escape ladder is 6.0 m long when extended to the icy alley below. It is held at the top by a frictionless pivot, and there is negligible frictional force from the icy surface at the bottom. The ladder weighs 250 N, and its center of mass (and gravity) is 2.0 m along the ladder from its bottom. A mother and child of total weight 750 N are on the ladder 1.5 m from the pivot. The ladder makes an angle θ with the horizontal.

a) Find the magnitude and direction of the force exerted by the icy alley on the ladder.

b) Find the magnitude and direction of the force exerted by the ladder on the pivot.

c) Do your answers in parts (a) and (b) depend on the angle θ?

S05. A uniform 250 kg beam is supported using a cable connected to the ceiling, as shown in the figure. The lower end of the beam rests on the floor.

a) What is the tension in the cable?

b) What is the minimum coefficient of static friction between the beam and the floor required for the beam to remain in this position?

S06. One end of a uniform meter stick is placed against a vertical wall as shown in the figure. The other end is held by a lightweight cord that makes an angle θ with the stick. The coefficient of static friction between the end of the meter stick and the wall is 0.40.

a) What is the maximum value the angle θ can have if the stick is to remain in equilibrium? (We aren’t hanging the block on it yet.)

b) Let the angle θ = 15o. A block of the same weight as the meter stick is suspended from the stick, as shown, at a distance x from the wall. What is the minimum value of x for which the stick will remain in equilibrium?

c) When θ = 15o, how large must the coefficient of static friction be so that the block can be attached 10 cm from the left end of the stick without causing it to slip?

I01. Consider the situation illustrated at right. Four point masses (each 1kg) are held together with stiff rods of negligible mass, forming four corners of a rectangle. Find the moment of inertia about the three axes shown (dotted lines).

I02. A thin rectangular plate of width a and height b has a mass M and is mounted on an axis that is parallel to its height as seen in the figure.

a) Show that the moment of inertia of this plate is (1/3)Ma2.

b) Find the moment of inertia of this plate about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the axis shown in the figure. (Use the parallel axis theorem for parts b and c.)

c) Find the moment of inertia of this plate about an axis that lies in the plane of the plate, passes through the center of the plate, and is perpendicular to the axis in part a.

I04. A uniform, very thin bar has two small balls glued to its ends. The bar is 2.00 m long with mass 4.00 kg, while the balls each have mass 0.500 kg and can be treated like point masses. Find the moment of inertia of this combination about each of the following axes:

a) an axis perpendicular to the bar through its center;

b) an axis perpendicular to the bar through one of the balls;

c) an axis parallel to the bar through both balls;

d) an axis parallel to the bar and 0.500 m from it.

I05. The three uniform objects shown in the diagram below have the same mass m. Object A is a solid cylinder with radius R. Object B is a hollow, thin cylinder with radius R. Object C is a solid cube whose sides are of length 2R. The axis of rotation of each object is perpendicular to the page and through the object’s center of mass.

a) Which object has the smallest moment of inertia? Explain.

b) Which object has the largest moment of inertia? Explain.

c) Where would the moment of inertia of a uniform solid sphere of mass m and radius R rank (axis through the center of mass)? Explain.

(over)