Chapter 1

The angles between the vector \(\mathbf{F}=F_{x} \mathbf{i}+F_{y} \mathbf{j}+F_{\mathbf{z}} \mathbf{k}\) and the coordinate axes are \(\theta_{x}, \theta_{y}\), and \(\theta_{z}\) The cosines of these angles are known as direction cosines. Evaluate the direction cosines in terms of the components of \(\mathbf{F}\). Show that $$ \cos ^{2} \theta_{x}+\cos ^{2} \theta_{y}+\cos ^{2} \theta_{z}=1 $$

According to the distributive law for vector cross products $$ \mathbf{r} \times \mathbf{F}_{1}+\mathbf{r} \times \mathbf{F}_{2}=\mathbf{r} \times\left(\mathbf{F}_{1}+\mathbf{F}_{2}\right) $$ This states that the sum of the moments of two concurrent forces about a point is equal to the moment of the vector sum of the forces about the same point. Verify this by simple geometry in the special case where \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) lie in the \(x y\) plane and intersect at the point \(P\), and \(\mathbf{r}\) is the displacement vector \(O P\).

According to the distributive law for vector cross products $$ \mathbf{r} \times \mathbf{F}_{1}+\mathbf{r} \times \mathbf{F}_{2}=\mathbf{r} \times\left(\mathbf{F}_{1}+\mathbf{F}_{2}\right) $$ This states that the sum of the moments of two concurrent forces about a point is equal to the moment of the vector sum of the forces about the same point. Verify this by simple geometry in the special case where \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) lie in the \(x y\) plane and intersect at the point \(P\), and \(\mathbf{r}\) is the displacement vector \(O P\).

Let \(\mathbf{F}\) be an arbitrary vector through the origin and let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be three arbitrary noncoplanar unit vectors passing through the origin. It is desired to decompose \(\mathbf{F}\) into vectors parallel to \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c} ;\) that is, it is desired to find magnitudes \(L_{a}, L_{b}\), and \(L_{c}\) such that $$ \mathbf{F}=L_{a} \mathbf{a}+L_{b} \mathbf{b}+L_{c} \mathbf{c} $$ Sketch this and show that it involves a parallelepiped whose edges are parallel to \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) with \(\mathbf{F}\) as the diagonal. Use the properties of the scalar triple product to show that $$ L_{a}=\frac{\mathbf{F} \times \mathbf{b} \cdot \mathbf{c}}{\mathbf{a} \times \mathbf{b} \cdot \mathbf{c}} $$ with similar expressions for \(L_{b}\) and \(L_{c^{\circ}}\)

The top of a tin can is removed, and the empty can is inverted over a pair of billiard balls on a table as shown in the sketch. For certain combinations of sizes and weights the configuration shown is stable. For other combinations the can tips over when released. It is proposed to set up a demonstration for a temperance lecture by using in sequence a frozen-orange-juice can and a beer can with the same pair of billiard balls. Investigate whether tipping will occur for the following sizes and weights. \(\begin{array}{lcl} & \text { Orange juice } & \text { Beer } \\ \text { Diameter of ball } & 45 \mathrm{~mm} & 45 \mathrm{~mm} \\ \text { Weight of ball } & 2.0 \mathrm{~N} & 2.0 \mathrm{~N} \\ \text { Diameter of can } & 50 \mathrm{~mm} & 70 \mathrm{~mm} \\ \text { Weight of empty can with lid removed } & 0.57 \mathrm{~N} & 1.0 \mathrm{~N}\end{array}\)

A rigid rod with negligible weight and small transverse dimensions carries a load \(W\) whose position is adjustable. The rod rests on a small roller at \(A\) and bears against the vertical wall at \(B\). Determine the distance \(x\) for any given value of \(\theta\) such that the rod will be in equilibrium. Assume that friction is negligible.

A freely pivoted light rod of length \(l\) is pressed against a rotating wheel by a force \(P\) applied to its middle. The friction coefficient between the rod and wheel materials is \(f\). Compute, for both directions of rotation, the friction force \(F\) as a function of the variables \(l, P\), and \(f\), and any others which are relevant. One of these two situations is sometimes referred to as a friction lock. Which one, and why?

The clean-air car shown has the following characteristics: Wheelbase \(L=250 \mathrm{~cm}\) Weight \(W=10 \mathrm{kN}\) Weight distribution (on level ground), 60 percent on rear wheels Power h.p. \(=75 \mathrm{MW}\), rear-wheel drive Height of center of gravity \(h=0.5 \mathrm{~m}\) Wheel diameter \(d=0.5 \mathrm{~m}\) If the coefficient of friction between tires and road is \(f=0.7\), what is the maximum hill angle \(\theta\) that can be climbed?

A circular cylinder \(A\) rests on top of two half-circular cylinders \(B\) and \(C\), all having the same radius \(r\) The weight of \(A\) is \(W\) and that of \(B\) and \(C\) is \(1 / 2 W\) each. Assume that the coefficient of friction between the flat surfaces of the halfcylinders and the horizontal table top is \(f\). Determine the maximum distance \(d\) between the centers of the half-cylinders to maintain equilibrium.

A block of weight \(W\) rests on an inclined plane which makes an angle \(\theta=\tan ^{-1} 3 / 4\) as shown. A force \(P\), parallel to the \(x\) axis, is applied to the block and gradually increased from zero; when \(P\) reaches the value \(0.4 W\) the block begins to slide. What is the coefficient of friction between the block and the inclined plane?

A longshoreman can barely start pushing a trunk up a \(30^{\circ}\) concrete ramp. He can barely hold it from sliding back when the slope is \(60^{\circ}\). What is the coefficient of static friction between the trunk and the concrete?

In building an orbiting space laboratory it will be necessary to drill holes in the flat steel wall of a space vehicle. The astronaut doing the drilling is unable to apply any appreciable amount either of force or of torque during the drilling so that it is necessary to mount the drill in a holder with three legs terminating in magnets, which grip the wall of the space vehicle. If the drilling torque is \(15 \mathrm{~N} \cdot \mathrm{m}\), and the normal force is \(50 \mathrm{~N}\), compute the minimum allowable holding force at each leg if the friction coefficient between the legs and space vehicle is \(0.4\).