Question

The trajectory of a particle moving on a straight line is x(t) = A cos ωt + B sin ωt. a) What are the units for the fixed numbers A, B and ω (the greek letter omega), assuming that x is measured in meters and t in seconds? b) There is a shortest non-zero time T such that x(t + T) = x(t); what is it? c) What is the velocity of the particle? d) What are the initial position x(0) and velocity vx(0) of the particle? e) The total force acting on the particle at time t, F total(t), equals max(t). Write the total force in terms of x(t) and the angular frequency ω.

Answer #1

For a linear oscillator (block on spring),
x(t) = xm cos (ωt + φ) (1)
v(t) = −ωxm sin (ωt + φ) (2)
a(t) = −ω
2xm cos (ωt + φ). (3)
(a) Draw and explain in words how you would use a circle diagram
to
connect x(t), v(t), and the phase (ωt + φ).
(b) What does ω represent? How is ω different for a linear
oscillator
than for a rolling or rotating object?

A particle is moving according to the given data a(t) = sin t +
3 cos t, x(0) = 0, v(0) = 2.
where a(t) represents the acceleration of the particle at time
t. Find v(t), the velocity of the particle at time t, and x(t), the
position of the particle at time t.

A particle of mass m, is under the influence of a force F given
by
F = Fo [(sin ωt)ˆi + (cos ωt) ˆj]
where F0, ω are positive constants. If at t = 0 the particle is
at rest at the origin, find
(a) the equations of motion x (t) and y (t) of the particle,
and
(b) the work done by the force F from t = 0 to t = 2π/ω.

Let a > b. Suppose a particle moves in an elliptical path
given by r(t) = (a cos ωt) i+(b sin ωt) j where ω > 0. Sketch
its velocity and acceleration vectors at one of the vertices of the
ellipse (±a, 0).

A particle of mass 10kg moves in a straight line such that the
force (in Newtons) acting on it at time (in seconds) is given by
90t4+70t3+30,
If at time t=0 its velocity,v (in ms-1), is given by
v(0)=9 , and its position x (in m) is given by x(0)=6 , what is the
position of the particle at time ?

The position of a particle moving with constant acceleration is
given by x(t) = 2t2 +
8t + 4 where x is in meters and t is in
seconds.
(a) Calculate the average velocity of this particle between
t = 6 seconds and t = 9 seconds.
(b) At what time during this interval is the average
velocity equal to the instantaneous velocity?

The position of a particle moving with constant acceleration is
given by
x(t) = 4t2 + 3t +
4
where x is in meters and t is in seconds.
(a) Calculate the average velocity of this particle between
t = 2 seconds and t = 7 seconds.
(b) At what time during this interval is the average velocity equal
to the instantaneous velocity?
(c) How does this time compare to the average time for this
interval?
a. It is larger....

A particle that moves along a straight line has velocity v ( t )
= t^2e^− 2t meters per second after t seconds. How many meters will
it travel during the first t seconds (from time=0 to time=t)?

The displacement (in centimeters) of a particle moving back and
forth along a straight line is given by the equation of motion
s = 4 sin(πt) + 5
cos(πt),
where t is measured in seconds. (Round your answers to
two decimal places.)
(a) Find the average velocity during each time period.
(i) [1, 2] cm/s
(ii) [1, 1.1]
cm/s
(iii) [1, 1.01]
m/s
(iv) [1, 1.001]
(b) Estimate the instantaneous velocity of the particle when
t = 1.

The displacement (in centimeters) of a particle moving back and
forth along a straight line is given by the equation of motion
s = 3 sin(πt) + 4
cos(πt),
where t is measured in seconds. (Round your answers to
two decimal places.)
(a) Find the average velocity during each time period.
(i) [1, 2]
cm/s
(ii) [1, 1.1]
cm/s
(iii) [1, 1.01]
cm/s
(iv) [1, 1.001]
cm/s
(b) Estimate the instantaneous velocity of the particle when
t = 1.
cm/s

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