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PH1104/CY1305: Mechanics

Semester 1 2013

Homework 2

1. A ball slides down the track (the track sits on the surface of the earth) from rest at A all the way to

B, with the side view as shown in the figure, and is always in contact with the track. (Ignore air

resistance and friction.)

A

B

There are two forces acting on the ball throughout its motion along the track. What are they?

What are the reaction pairs to each of the two forces?

Which of the two forces on its own would change the direction of motion of the ball? Explain.

Which of the two forces on its own would change the speed of the ball? Explain.

2. You are taking a stroll at a park, holding a balloon filled with helium gas. At a moment of

distraction, you accidentally let go of your grasp on the balloon (so its initial velocity is zero). The

amount of helium gas in the balloon is such that the buoyancy due to the air exactly supports its

weight (so it practically hovers at some fixed height).

It is a windy evening. The figure below shows the forces due to the wind at various points across

the park (top view), which happen not to be changing during that entire evening.

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Describe the trajectory of the helium balloon, in the two situations where it is released at point A

and at point B. In doing so, explain carefully how the forces due to the wind at the various

positions would cause the balloon to accelerate and affect its direction of motion.

Point C is a special point where the net force due to the wind is zero. Suppose you release the

balloon at point C, so that it is at equilibrium. Is it a stable equilibrium, i.e. would it remain at rest

if a bee happens to give it a slight nudge?

3. A small ball rolls around a horizontal circle at height y inside the cone shown below. Find an

expression for the ball’s speed in terms of a, h, y and g.

4. A kid with mass m wears roller skates whose coefficient of friction with the floor is μ. He ties a

massless rope around his waist, passes it around a frictionless pulley, and grabs hold of the other

end as shown in the figure. He then pulls it with constant force F. What is his acceleration

towards the wall?

In describing your solution, please clearly identify the forces acting on which object and caused

by which object.

5. A small bead can slide without friction on a circular hoop that is in a

vertical plane and has a radius of 0.100 m. The hoop rotates at a

constant rate of 4.00 rev/s about a vertical diameter (see figure).

(a) Find the angle β at which the bead is in vertical equilibrium. (Of

course, it has a radial acceleration toward the axis.)

(b) Is it possible for the bead to “ride” at the same elevation as the

center of the hoop?

(c) Show that the bead would ride at the bottom of the hoop if the

hoop rotates at 1.00 rev/s. Give a physical explanation to this.

(In this problem, do not use the notion of angular velocity. Consider

instead, the tangential velocity of the bead around the circular path

being the circumference of that circular path divided by the period of revolution.)