A particle starts from rest and travels in a circular path of diameter 10.0 m. If the tangential component of its acceleration is 2.5 m/s2, determine (a) the speed at the instant it has traveled one third of a lap around the circle, and (b) the magnitude of the total acceleration at that instant.
To solve this problem, we need to use the equations for circular motion.
(a) To find the speed at the instant the particle has traveled one third of a lap around the circle, we can use the formula for the circumference of a circle, C = πd, where d is the diameter.
Given that the diameter is 10.0 m, the circumference of the circle is C = π * 10.0 m = 31.4 m.
Since the particle has traveled one third of the lap, it has traveled a distance of 1/3 * 31.4 m = 10.47 m.
To find the speed, we can use the formula for linear speed: v = s/t, where v is the speed, s is the distance traveled, and t is the time taken.
Since the particle starts from rest, its initial speed is 0 m/s. Therefore, the final speed is equal to the average speed.
Assuming the time taken to travel 10.47 m is t, we can rearrange the formula to solve for v:
v = s / t
v = 10.47 m / t
(b) To find the magnitude of the total acceleration at that instant, we need to find both the tangential acceleration and the radial acceleration.
The tangential acceleration can be determined using the given value of 2.5 m/s^2.
The radial acceleration can be found using the formula for centripetal acceleration: ar = v^2 / r, where ar is the radial acceleration, v is the speed, and r is the radius of the circle (half the diameter).
Given that the diameter is 10.0 m, the radius is 5.0 m.
Substituting the values, we can find the radial acceleration:
ar = v^2 / r
ar = v^2 / 5.0 m
Now, we can proceed to find the final speed and the magnitude of the total acceleration.
To solve for the final speed and the total acceleration, we need to calculate the time taken to travel the given distance.
Since we know the initial speed is 0 m/s, and the acceleration is tangential, we can use the equation v = u + at, where u is the initial speed, v is the final speed, a is the acceleration, and t is the time.
Since u = 0 m/s and a = 2.5 m/s^2, we can rearrange the equation to solve for t:
t = v / a
t = 10.47 m / 2.5 m/s^2
Now, we have all the values needed to find the final speed and the magnitude of the total acceleration.
To calculate the final speed, use v = 10.47 m / t.
To calculate the total acceleration, add the tangential acceleration and the radial acceleration:
atotal = √((at)^2 + (ar)^2)
Now, plug in the values and calculate the final speed and the magnitude of the total acceleration.