A ball rolls on a circular track of radius 0.65 m with a constant angular speed of 1.2 rad/s in the counterclockwise direction.
Part A
If the angular position of the ball at t= 0 is theta= 0, find the x component of the ball's position at the time 2.6 s. Let theta= 0 correspond to the positive x direction.
Part B
Find the x component of the ball's position at the time 5.1 s.
Part C
Find the x component of the ball's position at the time 7.6 s.
Part A:
Well, the ball is rolling counterclockwise on the circular track, so its x position will be given by x = r * cos(theta), where r is the radius of the track and theta is the angular position of the ball at any given time. Since theta = 1.2 * t, we can substitute that into the equation to get x = 0.65 * cos(1.2 * t).
Plugging in t = 2.6 s, we have x = 0.65 * cos(1.2 * 2.6). Go ahead and calculate that to find the x component of the ball's position.
Part B:
Continuing with the same equation x = 0.65 * cos(1.2 * t), we just plug in t = 5.1 s to find the x component of the ball's position at that time.
Part C:
Again, using x = 0.65 * cos(1.2 * t), substitute t = 7.6 s to find the x component of the ball's position.
why we use cos instead of sin?
Part A:
To find the x component of the ball's position at time t = 2.6 s, we can use the formula for circular motion:
x = r * cos(theta)
Given that the radius of the track is 0.65 m and the angular position at t = 0 is theta = 0, we can substitute these values into the equation:
x = 0.65 * cos(1.2 * 2.6)
Simplifying the equation gives:
x = 0.65 * cos(3.12)
Using a calculator to find the value of cos(3.12), we get:
x ≈ 0.65 * (-0.999).
Thus, the x component of the ball's position at t = 2.6 s is approximately -0.6475 meters.
Part B:
To find the x component of the ball's position at time t = 5.1 s, we once again use the formula for circular motion:
x = r * cos(theta)
With the same radius of 0.65 m and assuming theta is still 0 at t = 0, we have:
x = 0.65 * cos(1.2 * 5.1)
Simplifying the equation gives:
x = 0.65 * cos(6.12)
Using a calculator to find the value of cos(6.12), we get:
x ≈ 0.65 * (0.988).
Thus, the x component of the ball's position at t = 5.1 s is approximately 0.6412 meters.
Part C:
Finally, to find the x component of the ball's position at time t = 7.6 s, we use the formula for circular motion:
x = r * cos(theta)
With the same radius of 0.65 m and assuming theta is still 0 at t = 0, we have:
x = 0.65 * cos(1.2 * 7.6)
Simplifying the equation gives:
x = 0.65 * cos(9.12)
Using a calculator to find the value of cos(9.12), we get:
x ≈ 0.65 * (-0.983).
Thus, the x component of the ball's position at t = 7.6 s is approximately -0.63795 meters.
To solve this problem, we need to use the equations of motion for circular motion. Let's start by calculating the angular displacement of the ball.
Part A:
Given that the angular speed of the ball is 1.2 rad/s and the time is 2.6 s, we can calculate the angular displacement using the formula:
θ = ω * t
where θ is the angular displacement, ω is the angular speed, and t is the time.
θ = 1.2 rad/s * 2.6 s
θ = 3.12 rad
The angular position is given by θ = 0 at t = 0, which corresponds to the positive x direction. Therefore, the initial position of the ball is on the positive x-axis.
Now, let's calculate the x component of the ball's position at the given time.
The x component of the ball's position can be found using the formula:
x = r * cos(θ)
where x is the x component of the position, r is the radius of the circular track, and θ is the angular position.
Substituting the given values, we have:
x = 0.65 m * cos(3.12 rad)
x ≈ 0.65 m * (-0.9962)
x ≈ -0.6477 m
Therefore, the x component of the ball's position at the time 2.6 s is approximately -0.6477 m.
Part B and Part C:
We can apply the same steps to determine the x component of the ball's position at different times.
For Part B, where t = 5.1 s:
θ = 1.2 rad/s * 5.1 s
θ ≈ 6.12 rad
x = 0.65 m * cos(6.12 rad)
x ≈ 0.65 m * (-0.7961)
x ≈ -0.5177 m
Therefore, the x component of the ball's position at the time 5.1 s is approximately -0.5177 m.
For Part C, where t = 7.6 s:
θ = 1.2 rad/s * 7.6 s
θ ≈ 9.12 rad
x = 0.65 m * cos(9.12 rad)
x ≈ 0.65 m * (0.2344)
x ≈ 0.1521 m
Therefore, the x component of the ball's position at the time 7.6 s is approximately 0.1521 m.
x=Acos(wt)
x= position
A=amplitude (in a circle radius is always amplitude)
w= angular frequency
t= time(s)
step1: convert rad-> degrees
w=1.2 radians*57.29 (constant)= 68.75degrees
step2: x=Acos(wt)
x=.65m*cos(68.75degrees*time)
just substitute time into equation above