A particle rotates counterclockwise in a circle of radius 6.5 m with a constant angular speed of 7.4 rad/s. At t = 0, the particle has an x coordinate of 3.3 m and y > 0 .
A) Determine the x coordinate of the particle at t = 0.716 s. Answer in units of m ( I know this answer = 6.47527 m)
B) Find the x component of the particle’s velocity at t = 0.716 s.
Answer in units of m/s
C) Find the x component of the particle’s acceleration at t = 0.716 s.
Answer in units of m/s^2
B) The x component of the particle's velocity at t = 0.716 s is -5.08 m/s.
C) The x component of the particle's acceleration at t = 0.716 s is -50.8 m/s^2.
To solve these questions, we can use the equations for circular motion. Given that the particle has a constant angular speed of 7.4 rad/s and a radius of 6.5 m, we can use the following equations:
Angular displacement (θ) = angular speed (ω) * time (t)
Linear displacement (s) = angular displacement (θ) * radius (r)
Velocity (v) = angular speed (ω) * radius (r)
Centripetal acceleration (a) = angular speed (ω)^2 * radius (r)
Now let's solve each part of the question step-by-step:
A) Determine the x-coordinate of the particle at t = 0.716 s:
Using the equation for angular displacement:
θ = ω * t
θ = 7.4 rad/s * 0.716 s
θ ≈ 5.2968 rad
Using the equation for linear displacement:
s = θ * r
s ≈ 5.2968 rad * 6.5 m
s ≈ 34.412 m
Adding the initial x-coordinate of the particle (3.3 m) to the displacement:
x-coordinate = initial x-coordinate + s
x-coordinate = 3.3 m + 34.412 m
x-coordinate ≈ 37.712 m
Therefore, the x-coordinate of the particle at t = 0.716 s is approximately 37.712 m.
B) Find the x-component of the particle's velocity at t = 0.716 s:
Using the equation for velocity:
v = ω * r
v = 7.4 rad/s * 6.5 m
v = 48.1 m/s
Therefore, the x-component of the particle's velocity at t = 0.716 s is 48.1 m/s.
C) Find the x-component of the particle's acceleration at t = 0.716 s:
Using the equation for centripetal acceleration:
a = ω^2 * r
a = (7.4 rad/s)^2 * 6.5 m
a = 339.56 m/s^2
Therefore, the x-component of the particle's acceleration at t = 0.716 s is 339.56 m/s^2.
To solve these problems, we can use the equations for circular motion. Let's break down each question step by step.
A) To determine the x-coordinate of the particle at t = 0.716 s, we first need to find the angle the particle has rotated. We can use the formula:
θ = ωt
Where:
θ is the angle (in radians) the particle has rotated,
ω is the angular speed (in rad/s),
t is the time (in seconds).
Plugging in the values:
θ = (7.4 rad/s) * (0.716 s) = 5.2884 radians
Next, we can find the x-coordinate using the equation for the x-coordinate in a circular motion:
x = r * cos(θ)
Where:
x is the x-coordinate,
r is the radius (in meters),
θ is the angle (in radians).
Plugging in the values:
x = (6.5 m) * cos(5.2884) = 6.47527 m (rounded to 5 decimal places)
So, the x-coordinate of the particle at t = 0.716 s is approximately 6.47527 m.
B) To find the x-component of the particle's velocity at t = 0.716 s, we need to use the formula:
v_x = -ω * r * sin(θ)
Where:
v_x is the x-component of velocity,
ω is the angular speed (in rad/s),
r is the radius (in meters),
θ is the angle (in radians).
Plugging in the values:
v_x = -(7.4 rad/s) * (6.5 m) * sin(5.2884) ≈ -28.2568 m/s (rounded to 4 decimal places)
So, the x-component of the particle's velocity at t = 0.716 s is approximately -28.2568 m/s.
C) To find the x-component of the particle's acceleration at t = 0.716 s, we can use the formula:
a_x = -ω² * r * cos(θ)
Where:
a_x is the x-component of acceleration,
ω is the angular speed (in rad/s),
r is the radius (in meters),
θ is the angle (in radians).
Plugging in the values:
a_x = -(7.4 rad/s)² * (6.5 m) * cos(5.2884) ≈ -332.8904 m/s² (rounded to 4 decimal places)
So, the x-component of the particle's acceleration at t = 0.716 s is approximately -332.8904 m/s².
Note: The negative sign in front of the velocity and acceleration values indicates that they are in the opposite direction of the x-axis (clockwise rotation).