A particle is moving clockwise in a circle of radius 1.75 m. At a certain instant, the magnitude of its acceleration is

a =

a

= 39.0 m/s2,
and the acceleration vector has an angle of θ = 50° with the position vector, as shown in the figure. At this instant, find the speed,
v =

v

,
of this particle.

v= sqrt (a*r*cos 50)

To find the speed of the particle, we can use the relationship between acceleration, speed, and radius in circular motion.

The acceleration vector has two components: tangential acceleration (at) and centripetal acceleration (ac). The tangential acceleration is responsible for changing the magnitude of the particle's velocity, while the centripetal acceleration keeps the particle moving in a circular path.

Given that the magnitude of the acceleration is 39.0 m/s^2 and the angle between the acceleration vector and the position vector is 50°, we can find the tangential acceleration using the formula:

at = a * cos(θ)

where:
at = tangential acceleration
a = magnitude of acceleration
θ = angle between acceleration and position vector

Substituting the given values:
at = 39.0 * cos(50°)

Now, let's calculate the tangential acceleration:

at = 39.0 * cos(50°)
at = 39.0 * 0.6428
at ≈ 25.12 m/s^2

The tangential acceleration represents the rate at which the speed of the particle is changing. Since the acceleration is in the same direction as the velocity (clockwise), the speed of the particle is increasing.

To find the speed (v) at that instant, we need to find the initial speed (u) and the time taken (t), assuming the particle starts from rest:

Using the formula:
v = u + at * t

Since the particle starts from rest, u = 0. Therefore, the formula becomes:

v = at * t

We know that the centripetal acceleration (ac) is given by:

ac = v^2 / r

where:
ac = centripetal acceleration
v = speed
r = radius

Now, let's find the centripetal acceleration:

ac = v^2 / r
(ac * r) = v^2
v^2 = ac * r
v = √(ac * r)

Substituting the given values:
v = √(39.0 * 1.75)

Now, let's calculate the speed (v):

v = √(39.0 * 1.75)
v ≈ √(68.25)
v ≈ 8.26 m/s

Therefore, the speed of the particle at that instant is approximately 8.26 m/s.