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

Is the center of the circle the origin of x,y coordinate system?

To solve this problem, we can use the equations for circular motion. The x coordinate of a particle moving in a circle is given by x = r * cos(θ), where r is the radius and θ is the angle in radians.

A) To find the x coordinate at t = 0.716 s, we need to find the angle θ at this time. We can use the equation θ = ω * t, where ω is the angular speed and t is the time.

θ = 7.4 rad/s * 0.716 s
θ = 5.3016 radians

Now we can use the equation x = r * cos(θ) to find the x coordinate.

x = 6.5 m * cos(5.3016 radians)
x ≈ 6.47527 m

So, the x coordinate of the particle at t = 0.716 s is approximately 6.47527 m.

B) The x component of the particle's velocity can be found using the equation v_x = ω * r * sin(θ), where v_x is the x component of the velocity.

v_x = 7.4 rad/s * 6.5 m * sin(5.3016 radians)
v_x ≈ 29.442 m/s

Therefore, the x component of the particle's velocity at t = 0.716 s is approximately 29.442 m/s.

C) The x component of the particle's acceleration is given by a_x = -ω^2 * r * cos(θ), where a_x is the x component of the acceleration.

a_x = - (7.4 rad/s)^2 * 6.5 m * cos(5.3016 radians)
a_x ≈ -112.491 m/s^2

So, the x component of the particle's acceleration at t = 0.716 s is approximately -112.491 m/s^2.

To solve this problem, we'll need to break it down into different parts.

A) To determine the x-coordinate of the particle at t = 0.716 s, we can use the equation for rotational motion:

x = r * cos(θ)

Where x is the x-coordinate, r is the radius of the circle, and θ is the angle in radians. In this case, the radius is given as 6.5 m, and the angular speed is given as 7.4 rad/s. We can find θ by multiplying the angular speed by time:

θ = angular speed * time

So, θ = 7.4 rad/s * 0.716 s = 5.2912 rad

Now, we can substitute the values into the equation:

x = 6.5 m * cos(5.2912 rad)

Calculating this expression, we find that x is approximately equal to 6.47527 m.

B) To find the x-component of the particle's velocity at t = 0.716 s, we can again use the equation for rotational motion:

v = r * ω * sin(θ)

Where v is the velocity, r is the radius, ω is the angular speed, and θ is the angle in radians. We already have the values for r and ω from the previous part, and we know θ from our calculations:

v = 6.5 m * 7.4 rad/s * sin(5.2912 rad)

Evaluating this expression, we find that the x-component of the particle's velocity is approximately 44.0146 m/s.

C) Finally, to find the x-component of the particle's acceleration at t = 0.716 s, we can use the equation for centripetal acceleration:

a = r * ω^2

Where a is the acceleration, r is the radius, and ω is the angular speed. We can substitute the values we already have:

a = 6.5 m * (7.4 rad/s)^2

Evaluating this expression, we find that the x-component of the particle's acceleration is approximately 352.36 m/s^2.