Angular velocity is a measure of how fast an object is rotating around a fixed axis. It is usually denoted by the symbol ω and is measured in radians per second (rad/s). In this question, we are given that Gear 1 rotates at 60 rpm clockwise (cw) when looked from the left. We need to find the angular velocity of the carrier and its direction so that Gear 4 rotates in the counter clockwise (ccw) direction at twice the angular velocity of Gear 1 when looked from the left.

Let's break down the problem into steps:

Step 1: Convert Gear 1's angular velocity to rad/s
We are given that Gear 1 rotates at 60 rpm clockwise (cw). To convert this to rad/s, we use the conversion factor of 1 revolution = 2π radians. Therefore, the angular velocity of Gear 1 (ω1) can be calculated as follows:
ω1 = (60 rpm) * (2π rad/1 min) * (1 min/60 s)
ω1 = 2π rad/s

Step 2: Determine the desired angular velocity of Gear 4
We are asked to find the angular velocity of Gear 4 when looked from the left. It should be twice the angular velocity of Gear 1. Therefore, the desired angular velocity of Gear 4 (ω4) can be calculated as follows:
ω4 = 2 * ω1
ω4 = 2 * 2π rad/s
ω4 = 4π rad/s

Step 3: Find the angular velocity of the carrier
The angular velocity of the carrier (ωc) can be calculated using the formula for gear systems:
ωc = (ω4 - ω3) / (1 - N3/Nc)
where ω3 is the angular velocity of Gear 3, N3 is the number of teeth on Gear 3, and Nc is the number of teeth on the carrier.

Since Gear 3 is not mentioned in the question, we can assume that it is fixed and does not rotate. Therefore, ω3 = 0. Additionally, since the direction of rotation is not specified for the carrier, we can assume it to be clockwise (cw) for simplicity.

Step 4: Plug in the values to calculate the angular velocity of the carrier
Using the formula for ωc, we can calculate the angular velocity of the carrier as follows:
ωc = (4π rad/s - 0) / (1 - N3/Nc)
Since the question does not provide specific values for the number of teeth on Gear 3 and the carrier, we cannot calculate ωc directly. However, we can determine the relationship between the angular velocities of Gear 1 and the carrier.

Step 5: Determine the relationship between ω1 and ωc
Since Gear 1 is meshed with the carrier, their angular velocities must be related. When two gears are in mesh, the following relationship holds:
N1/Nc = ωc/ω1
where N1 is the number of teeth on Gear 1.

We can rearrange this equation to solve for ωc:
ωc = (Nc/N1) * ω1

Step 6: Determine the angular velocity of the carrier using the relationship
Since Gear 4 rotates in the ccw direction at twice the angular velocity of Gear