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 at a given angle theta, we need to determine the position of point A on the cam.
a) Velocity in terms of polar coordinates:
In polar coordinates, we can use the formula for velocity:
v = r * ω
where:
v is the velocity,
r is the distance from the center of rotation (cam's center),
and ω (omega) is the angular velocity.
To find the distance r, we can use the concept of a right triangle formed by the bar, the cam's center, and point A:
r = √[(rc + h)^2 + d^2]
where:
rc is the radius of the cam,
h is the distance from the cam's center to the arm pinning point,
and d is the perpendicular distance between the cam's center and the triangle formed by the bar and the arm pinning point.
In this case, since theta = 40 degrees, we can calculate d using trigonometry:
d = (rc + h) * sin(theta)
Substituting the values into the equations, we can calculate the velocity v.
b) Velocity in terms of cartesian coordinates:
To find the velocity in terms of cartesian coordinates, we need to convert the coordinates from polar to cartesian. The transformation equations are:
x = r * cos(theta)
y = r * sin(theta)
Using these equations, we can calculate the position of point A on the cam at theta = 40 degrees. Then we can find the velocity by taking the derivative of the position vector with respect to time. The magnitude of the velocity can be found using the equation:
v = √(dx/dt)^2 + (dy/dt)^2
where dx/dt and dy/dt are the derivatives of x and y, respectively.
By differentiating the position equations, substituting the values, and calculating the derivatives, we can find the velocity in terms of cartesian coordinates.