Chapter 6

A force with magnitude \(F=a \sqrt{x}\) acts in the \(x\) -direction, where \(a=9.5 \mathrm{N} / \mathrm{m}^{1 / 2} .\) Calculate the work this force does as it acts on an object moving from (a) \(x=0\) to \(x=3.0 \mathrm{m} ;\) (b) \(3.0 \mathrm{m}\) to \(6.0 \mathrm{m}\) and (c) \(6.0 \mathrm{m}\) to \(9.0 \mathrm{m}\)

The force exerted by a rubber band is given approximately by $$F=F_{0}\left[\frac{L_{0}-x}{L_{0}}-\frac{L_{0}^{2}}{\left(L_{0}+x\right)^{2}}\right]$$ where \(L_{0}\) is the unstretched length, \(x\) is the stretch, and \(F_{0}\) is a constant. Find the work needed to stretch the rubber band a distance \(x\)

You put your little sister (mass \(m\) ) on a swing whose chains have length \(L\) and pull slowly back until the swing makes an angle \(\phi\) with the vertical. Show that the work you do is \(m g L(1-\cos \phi)\)

Two unknown elementary particles pass through a detection chamber. If they have the same kinetic energy and their mass ratio is \(4: 1,\) what's the ratio of their speeds?

A tractor tows a plane from its airport gate, doing 8.7 MJ of work. The link from the plane to the tractor makes a \(22^{\circ}\) angle with the plane's motion, and the tension in the link is \(0.41 \mathrm{MN}\). How far does the tractor move the plane?

A force pointing in the \(x\) -direction is given by \(F=F_{0}\left(x / x_{0}\right)\) where \(F_{0}\) and \(x_{0}\) are constants and \(x\) is position. Find an expression for the work done by this force as it acts on an object moving from \(x=0\) to \(x=x_{0}\)

A force pointing in the \(x\) -direction is given by \(F=a x^{3 / 2},\) where \(a=0.75 \mathrm{N} / \mathrm{m}^{3 / 2} .\) Find the work done by this force as it acts on an object moving from \(x=0\) to \(x=14 \mathrm{m}\).

Two vectors have equal magnitude, and their scalar product is one third the square of their magnitude. Find the angle between them.

By measuring oxygen uptake, sports physiologists have found that long-distance runners' power output is given approximately by \(P=m(b v-c),\) where \(m\) and \(v\) are the runner's mass and speed, and \(b\) and \(c\) are constants given by \(b=4.27 \mathrm{J} / \mathrm{kg} \cdot \mathrm{m}\) and \(c=1.83\) W/kg. Determine the work done by a 54 -kg runner who runs a 10 -km race at \(5.2 \mathrm{m} / \mathrm{s}\)

You're writing performance specifications for a new car model. The 1750 -kg car delivers energy to its drive wheels at the rate of \(35 \mathrm{kW} .\) Neglecting air resistance, what do you list for the greatest speed at which it can climb a \(4.5^{\circ}\) slope?