PH-UY 1013 Lecture Notes - Lecture 25: Horse Length

A 380 g block connected to a light spring for which the forceconstant is 8.90 N/m is free to oscillate on a horizontal,frictionless surface. The block is displaced 5.30 cm fromequilibrium and released from rest as in the figure.

(a) Express the position, velocity, and acceleration as functionsof time.

Find the phase constant from the initial condition that x = A at t= 0:
x(0) = A cosϕ = A --> ϕ = 0

Use the equation to write an expression for x(t):
x = Acos(ωt + ϕ) = ?

Use the equation to write an expression for v(t):
v = -ωAsin(ωt + ϕ) = ?

Use the equation to write an expression for a(t):
a = -ω^2 Acos(ωt + ϕ) = ?

A particle of mass m moves under the influence of a central force.The potential energy function is
V(r) = {
K (a2−r2) /2 if r ≤ a
0 if r > a

as illustrated in the figure. The initial position is (x,y) = (a/2,0) and the initial velocity is (vx,vy) = (0, v0).
[Data: m = 1.0 kg; K =17.1 N/m; a = 1.0 m; r0 = 0.5 a; v0 = 2.0m/s.]

(A) Determine the time ta when the mass crosses the boundary r =a.

(B) Determine angular position φa where the particle crosses theboundary r = a.

***HINT GIVEN***

Use the constants of the motion to analyze the orbit.

Write L = mr^2 dφ/dt = the initial value.
What is the initial value?

Write E = (1/2)m (dr/dt)^2 + L^2/(2mr2) +V(r )
= the initial value. What is the
initial value?

Part A. In the energy equation,
change the variable from r to u,
where r =sqrt(u). The result can be
solved by u(t) = c1+c2 cosh(αt).
Determine c1, c2, and α.

Part B.
φ = Integral 0 to t [L/(mr^2)] dt

particleof mass m = 1.18 kg is attached between two identical springs on ahorizontal frictionless tabletop. The springs have spring constantk, and each is initially unstressed. (a) If the mass of pulled adistance x along a direction perpendicular to the initialconfiguration of the springs, show that the potential energy of thesystem is:
U(x)=K(X^2) + 2KL (L- square-root(X^2 + L^2) {look above thequestion for better viewing equation }
(b) Make a plot of U(x) versus x and identify all equilibriumpoints.
Assume that L = 1.20 m and k = 40.0 N/m. (c) If the mass is
pulled 0.500 to the right and then released, what is its speed whenit
reaches the equilibrium point x = 0? (d) Using the potentialenergy
of the system find the force as a function of x.

NOTE: the picture is top view pictute