1. Newton’s Laws and damped simple harmonic motion A particle of mass m = 5 moves...
1. Newton’s Laws and damped simple harmonic motion
A particle of mass m = 5 moves in a straight line on a horizontal
surface. It is subject
to the following forces:
-
an attractive force in the direction of the fixed origin O with magnitude 40 times
the instantaneous distance from O
-
a damping force due to friction which is 20 times the instantaneous speed
-
the force due to gravity
-
the normal force.
The particle starts from rest at a distance of 20m from O.
-
(a) Use Newton’s laws to find the equations of motion and the magnitude of the
normal force.
-
(b) Write down the initial conditions and determine the position of the particle at
time t.
-
(c) Find the speed and velocity of the particle at time t.
-
(d) Find the time taken for the particle to reach the origin for the first time. State at what time it will return to the origin and describe the behaviour of the particle as t becomes large.
Homework Answers
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1. Newton’s Laws and damped simple harmonic motion
A particle of mass m = 5 moves...
A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is...
A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is 8.0 x 10-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum displacement, x = 1.5 m, at time t = 0, what is the angular frequency of the oscillations?
A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is...
A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is 8.0 x 10-3 kg·m/s, and the mass is 0.050 kg. If the particle starts at its maximum displacement, x = 1.5 m, at time t = 0, what is the particle’s position at t = 5.0 s? -1.5 m -0.73 m 0 m 0.73 m 1.5 m
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position,...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 3.50cm, and the frequency is 2.30 Hz. (a) Find an expression for the position of the particle as a function of time. (Use the following as necessary: t. Assume that x is in centimeters and t is in seconds. Do not include units in your answer.) x...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position,...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is2.50 cm, and the frequency is 1.30 Hz. (a) Find an expression for the position of the particle as a function of time. (Use the following as necessary: t, and ?.) x = (b) Determine the maximum speed of the particle. cm/s (c) Determine the earliest time (t...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position,...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t0 and moves to the right. The amplitude of its motion is 2.50 cm, and the frequency is 2.90 Hz. (a) Find an expression for the position of the particle as a function of time. (Use the following as necessary: t. Assume that x is in centimeters and t is in seconds. Do not include units in your answer.) x2.5sin (5.8xt)...
A particle undergoes uniform circular motion. This means that it moves in a circle of radius...
A particle undergoes uniform circular motion. This means that it moves in a circle of radius R about the origin at a constant speed. The position vector of this motion can be written Here, analogous to the simple harmonic motion problem of HW 1, ω is the angular frequency and has units of rad/s 1/s and can also be written in terms of the period of the motion as 2π (a) Show that the particle resides a distance R away...
A particle moves in simple harmonic motion with a frequency of 3.80 Hz and an amplitude...
A particle moves in simple harmonic motion with a frequency of 3.80 Hz and an amplitude of 5.50 cm. (a) Through what total distance does the particle move during one cycle of its motion? cm (b) What is its maximum speed? cm/s Where does this maximum speed occur? as the particle passes through equilibrium at maximum excursion from equilibrium exactly halfway between equilibrium and maximum excursion none of these (c) Find the maximum acceleration of the particle. m/s^2 Where in...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position,...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.60 cm, and the frequency is 1.20 Hz (a) Find an expression for the position of the particle as a function of time. (Use the following as necessary: t. Assume that x is in centimeters and t is in seconds. Do not include units in your answer.)...
(1 point) This problem is an example of critically damped harmonic motion. A mass m =...
(1 point) This problem is an example of critically damped harmonic motion. A mass m = 6 kg is attached to both a spring with spring constant k = 150 N/m and a dash-pot with damping constant c = 60 N· s/m . The ball is started in motion with initial position Xo = 8 m and initial velocity vo = -42 m/s. Determine the position function x(t) in meters. x(t) = Graph the function x(t). Now assume the mass...
3. A particle moving along the x axis in simple harmonic motion starts from its equilibrium...
3. A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t=0 s and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Determine the position, velocity, and acceleration equations for this particle. (b) Determine the maximum speed of this particle and the first time it reaches this speed after t=0 s.
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t0 and moves to the right. The amplitude of its motion is 2.50 cm, and the frequency is 2.90 Hz. (a) Find an expression for the position of the particle as a function of time. (Use the following as necessary: t. Assume that x is in centimeters and t is in seconds. Do not include units in your answer.) x2.5sin (5.8xt)...
A particle undergoes uniform circular motion. This means that it moves in a circle of radius R about the origin at a constant speed. The position vector of this motion can be written Here, analogous to the simple harmonic motion problem of HW 1, ω is the angular frequency and has units of rad/s 1/s and can also be written in terms of the period of the motion as 2π (a) Show that the particle resides a distance R away...
A particle moves in simple harmonic motion with a frequency of 3.80 Hz and an amplitude of 5.50 cm. (a) Through what total distance does the particle move during one cycle of its motion? cm (b) What is its maximum speed? cm/s Where does this maximum speed occur? as the particle passes through equilibrium at maximum excursion from equilibrium exactly halfway between equilibrium and maximum excursion none of these (c) Find the maximum acceleration of the particle. m/s^2 Where in...
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.60 cm, and the frequency is 1.20 Hz (a) Find an expression for the position of the particle as a function of time. (Use the following as necessary: t. Assume that x is in centimeters and t is in seconds. Do not include units in your answer.)...
(1 point) This problem is an example of critically damped harmonic motion. A mass m = 6 kg is attached to both a spring with spring constant k = 150 N/m and a dash-pot with damping constant c = 60 N· s/m . The ball is started in motion with initial position Xo = 8 m and initial velocity vo = -42 m/s. Determine the position function x(t) in meters. x(t) = Graph the function x(t). Now assume the mass...
3. A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t=0 s and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Determine the position, velocity, and acceleration equations for this particle. (b) Determine the maximum speed of this particle and the first time it reaches this speed after t=0 s.
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