Angular velocity of gears

In summary, we have gear C with a radius of 0.12 m rotating at an angular velocity of 1.9 k rad/s and a connecting link rotating at an angular velocity of 1.5 k rad/s. Gear D, with a radius of 0.05 m, is a planetary gear and its angular velocity needs to be found. The gear ratio ωc/ωd=rd/rc does not apply here. The correct solution is ωd=ωc x rcd / rd, resulting in an angular velocity of -2.125 rad/s. This is found by equating the velocities of points c and d and using unit vector ##\hat{t}## tangent to the gears at the
  • #1
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Gear C has radius 0.12 m and rotates with angular velocity 1.9 k rad/s. The connecting link rotates at angular velocity 1.5 k rad/s. Gear D has radius 0.05 m. Find the angular velocity of gear D (in rad/s). Note that gear C is pinned to ground and gear D is a planetary gear.

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Since gear c is pinned, the gear ratio ωc/ωd=rd/rc does not directly apply here right? I wonder if the correct solution would be ωd=ωc x rcd / rd. I got -2.125 rad/s. Thanks.
 
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  • #2
Hello.

At the point of contact of the gears, suppose you pick a point ##c## on the rim of gear C and a point ##d## on the rim of gear D as shown.

Let ##\vec{V_c}## and ##\vec{V_d}## be the velocities of those points, respectively, as measured relative to the fixed point at the center of gear C.

How are ##\vec{V_c}## and ##\vec{V_d}## related?
 

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  • #3
Vc and Vd are the same
 
  • #4
OK. So, ##\vec{V_c}## = ##\vec{V_d}##. If you can express each of these velocities in terms of the angular velocities of the gears (##\omega_C## and ##\omega_D##) and the angular velocity of the link (##\omega_L##), then you would have an equation that you could solve for ##\omega_D##.

Start with the velocity vector ##\vec{V_c}##. Suppose you introduce a unit vector ##\hat{t}## that is tangent to the gears at the point of contact, as shown. How can you express ##\vec{V_c}## in terms of ##\omega_C##, ##r_C##, and ##\hat{t}##?
 

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  • #5


I can confirm that your approach to finding the angular velocity of gear D is correct. The gear ratio formula ωc/ωd=rd/rc does not apply in this case since gear C is pinned and gear D is a planetary gear. Your solution of ωd=ωc x rcd / rd is the correct way to calculate the angular velocity of gear D. Plugging in the given values, we get ωd=(1.9 k rad/s) x (0.12 m) / (0.05 m) = 4.56 k rad/s or approximately 2.125 rad/s. This angular velocity represents the rate at which gear D is rotating, taking into account its smaller radius and the rotational speed of the connecting link.
 

1. What is angular velocity?

Angular velocity is a measure of the rate at which an object rotates around a fixed point. It is typically measured in radians per second or degrees per second.

2. How is angular velocity different from linear velocity?

Angular velocity measures the rotational speed of an object, while linear velocity measures the speed at which an object moves in a straight line. Angular velocity is dependent on the radius of rotation, while linear velocity is not.

3. How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angle by the change in time. It can also be calculated by dividing linear velocity by the radius of rotation.

4. How does the number of teeth on a gear affect its angular velocity?

The number of teeth on a gear does not directly affect its angular velocity. However, the size and shape of the teeth can impact the gear's ability to transfer angular velocity to other gears in a system.

5. Can angular velocity be negative?

Yes, angular velocity can be negative. A negative angular velocity indicates that the object is rotating in the opposite direction of a positive angular velocity.

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