This deals with a cam follower. The slotted bar rotates with constant angular velocity, theta=12 rad/s. The follower 'A' follows the circular cam as the bar rotates. The arm is pinned at a distance h=0.2m to the left of the cam's center. The radius of the cam is rc=0.42m.

Find the velocity of the follower when theta=40 degrees
a) in terms of polar coordinates
b) in terms of cartesian coordinates

To find the velocity of the follower when θ=40 degrees, we need to calculate both the radial velocity (dr/dθ) and the angular velocity (dθ/dt) at that point.

a) In terms of polar coordinates:
The radial velocity (dr/dθ) can be found using the equation:

dr/dθ = r * (dθ/dt)

To begin, we need to convert the given values to the corresponding dimensions in polar coordinates. The arm is pinned at a distance h=0.2m to the left of the cam's center, so the position of the follower in polar coordinates would be:

r = rc - h
= 0.42m - 0.2m
= 0.22m

Now, we need to find the angular velocity (dθ/dt) at that point. The given constant angular velocity is θ=12 rad/s. Since ω = dθ/dt, the angular velocity is already given.

Next, we can substitute the values into the equation:

dr/dθ = r * (dθ/dt)
dr/dθ = 0.22m * 12 rad/s
dr/dθ = 2.64 m/s

Therefore, the velocity of the follower when θ=40 degrees in terms of polar coordinates is 2.64 m/s.

b) In terms of cartesian coordinates:
To find the velocity of the follower in terms of cartesian coordinates (dx/dt, dy/dt), we need to convert the velocity from polar coordinates (dr/dθ, dθ/dt) to cartesian coordinates.

The conversion equations are:
dx/dt = (dr/dθ) * cos(θ) - r * sin(θ) * (dθ/dt)
dy/dt = (dr/dθ) * sin(θ) + r * cos(θ) * (dθ/dt)

First, convert the angle θ from degrees to radians:
θ = 40° * (π/180°)
= 40π/180
= 2π/9 rad

Substitute the given values:
dr/dθ = 2.64 m/s
r = 0.22m
dθ/dt = 12 rad/s
θ = 2π/9 rad

Now, substitute these values into the conversion equations:

dx/dt = (2.64 m/s) * cos(2π/9 rad) - (0.22m) * sin(2π/9 rad) * (12 rad/s)
dy/dt = (2.64 m/s) * sin(2π/9 rad) + (0.22m) * cos(2π/9 rad) * (12 rad/s)

Use a calculator to compute these values:

dx/dt ≈ -0.9864 m/s
dy/dt ≈ -1.4428 m/s

Therefore, the velocity of the follower when θ=40 degrees in terms of cartesian coordinates is approximately (-0.9864 m/s, -1.4428 m/s).