AP PHYSICS C – Mechanics 1987
AP PHYSICS C – Mechanics 1987
1. The adult's horizontal force is represented by FP.
[pic]
When the adult releases the swing, the swing has a 'v' or speed of 0 m/s. This is important.
It means that centripetal acceleration is zero: [pic]
We can now resolve the force of gravity or weight (w) into components as in the Inclined Plane Theorem. Fc = 0 means: [pic]
and F = ma (tangential direction): [pic] (mentioned here just to complete things!)
(Recall the theorem: [pic])
Back to the exam...
(d) At it's lowest point: [pic] Since W=mg, replace 'm' with: [pic]...
which gives us: [pic] but how do we replace v2 ?
Conservation of Mechanical Energy (E)
E0 = Ef
mgh = ½ mv2
[pic] Now substitute for 'h'
[pic](See diagram at right.)
v2 = [pic]
Finally, we can plug back into our Fc = mac equation.
[pic] and solve for T to get:
[pic]
AP PHYSICS C – Mechanics 1987
2. Once again we're given a potential energy graph! Remember: [pic]
[pic]
Now suppose the total energy (E) is constant and only 4.0 joules. (Note: U can't be 8J.)
(b) E = 4J = K + U
(i) So when x = 2m, U = 1J and [pic]
(ii) When x = 4m, U = 3J and [pic]
(c) At x = 0.5m, U = 6J and K ≥ 0J, so E > 4J. [pic]
(d) At x = 5m, U = 3.9J (less than 4J) ≤ E = 4J. [pic]
(e) Use [pic]. Remember to take the negative slope of the graph!
[pic]
Note the horizontal lines at F = 4N, then F = -1N, and finally at F = 2N.
AP PHYSICS C – Mechanics 1987
3. Linear momentum is not conserved! Hmm... Why? The hinge exerts an external force on the system. Notice that the hinge force exerts no torque on the system and hence angular momentum is conserved. K.E. is conserved (elastic collision)! Okay... let's go...
(a) Using conservation of kinetic energy (KT as well as KR):
½ (1)102 = ½ (1)v2 + [pic] and Ibar = [pic]
50 – ½ v2 = [pic] = [pic] Hmm... (didn't work!) we don't have v2, yet!
Let's switch to the swinging motion and conservation of mechanical energy (E):
E0 = Ef = Eff
50 = ½ v2 + [pic]= ½ v2 + mgh (30(.6) = 18J, since the center of mass is half-way)
[pic]= 18 [pic]
(b) Now we can find 'v':
50 = ½ v2 + mgh = ½ v2 + 18, hence v2 = 64 and [pic]
(c) [pic]
(d) We haven't used conservation of angular momentum yet!
L0 = Lf
12 = [pic] (We'll resolve the velocity to find a [pic] component, [pic])
12 = 1.44(5) + 1.2(1)[pic] = .72 + 1.2(8)[pic]
12 = 7.2 + 9.6[pic]
[pic]
-----------------------
In equilibrium: [pic]
Using our old trick of dividing the equations...
(b) [pic]
or solving for T-!QRSTUVYµ¶çèÿ [pic] [?] n o | } ” • – — ¼ ½ Ô Õ Ö × /0úõñíèíñÚÒíÎíÎÆηªÆÎ¥ÎÆΖ‰ÆÎÆÎzmÆÎÆÎ^jà±sB[pic]hRg7CJU[pic]V[pic]aJjQhRg7hRg7EHôÿU[pic]jb±sB[pic]hRg7CJU[pic]V[pic]aJj-
hRg7hRg7EHôÿU[pic]j›±sB[pic]hRg7CJU[pic]V[pic]aJ hRg7H*[?]jŸhRg7hR in the second equation we get...
(a) [pic]
[pic]
(a) Equilibrium points are
where F(x) = 0. Notice the slope of U(x) is zero when:
[pic]
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