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(PHYS111)[1999](f)midterm2~=tkk933^_27669.pdf
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Physics 111 2nd Midterm, 9 November, 1999 Time allowed: 90 minutes Total: 60 marks
Answer all questions. Take g = 9.8 ms .2 in your calculations. Answers should be given to 3 significant figures.
1. As shown in the figure, cart A has a mass of 2 kg and moves with a speed of 10 ms-1. Cart B has a mass of 3 kg and is initially stationary. Furthermore, cart B is equipped with a massless spring bumper of spring constant k = 2000 Nm-1.
(a)
When cart A collides elastically with cart B, what are the final velocities of carts A and B?

(b)
In this collision process, the spring is compressed to its maximum extent in half a second. Noting that the carts are moving with the same velocity at this instant, find the maximum compression of the spring during the collision? [10]


2. A uniform disc has a mass M and radius R. When it rotates about its central axis,
the rotational inertia is given by
1

I . MR2.
2 As shown in the figure, a uniform disc with mass M = 3 kg and radius R = 0.4 m is free to rotate in a vertical plane about a horizontal axis through its centre. A massless and inextensible string is wound around the disk and a block of mass m = 2 kg is hung down from it.
(a) What is the acceleration of the block?

3. The rotational inertia of a uniform rod about its centre is ML2/12, where M and L are its mass and length respectively.
(a)
As shown in the figure, what is the rotational inertia about an axis through a point P which is at a distance L/4 from one of its ends?

(b)
Suppose the rod is at rest in a horizontal plane. A mud ball, with mass M/2 and horizontal velocity v, hits the rod at its further end making an angle of 30o with the rod. The mud ball is then deformed and sticks at the rod, and the combination moves together. What is the angular velocity of the combination after the impact? [10]


1


4. Consider a comet travelling around the Sun in an ellipse. Its nearest and furthest distances from the Sun are re and 3re respectively, where re is the radius of the Earths orbit.
(a)
Using Keplers law of periods, find the period of the comets orbit.

(b)
Let v1 and v2 be the orbital velocities of the comet at the nearest and furthest points respectively. Using conservation of energy, find v12-v22 in units of v02, where


GM
v0 .
re

is the orbital velocity of the Earth around the Sun, with G being the universal gravitational constant, and M being the mass of the Sun.
(c) Using conservation of energy and conservation of angular momentum, find v1 and v2 in units of v0. [15]
5. A metal plate of mass 1 kg and uniform density on the xy-plane is bounded by a flat side defined by y=0 and the parabolic sides defined by x=.(4-y2) (length in meters).
(a)
What is the area A of the plate? [Hint: consider a thin slice with thickness dy at the height y. Find the area dA and integrate.]

(b)
What is the y-position of the centre of mass?

(c)
What is the rotational inertia of