The pulley in the figure represents different pulleys with outer radius and inner radius indicated in the table. The horizontal rope is pulled to the right at a constant speed that is the same in each case, and none of the ropes slips in its contact with the pulley
Rank these scenarios on the basis of the speed of the block.
R_outer: .4m
R_inner: .2m
R_outer: .6m
R_inner: .5m
R_outer: .4m
R_inner: .3m
R_outer: .6m
R_inner: .2m
R_outer: .2m
R_inner: .1m
Well, isn't this quite a pulley dilemma! Let's crank up the humor and have some pulley fun.
Alright, let's see which scenario takes the crown for the speediest block:
1) R_outer: .6m, R_inner: .2m: This is like a cosmic dance-off between an alien marshal and a hip-hop champion. With a big outer radius and a small inner radius, this scenario has the winner of the fastest block speed written all over it.
2) R_outer: .4m, R_inner: .2m: Ah, the classic battle of David and Goliath. In this scenario, the outer radius is smaller compared to the previous one, but the inner radius remains the same. It doesn't stand a chance against the previous scenario, but it still manages to sail through to second place.
3) R_outer: .4m, R_inner: .3m: Here we have a case of outer radius with a small reduction and a larger inner radius. It's like a penguin trying to do the floss dance. It's a valiant effort, but it can't match the speed of the previous two scenarios.
4) R_outer: .6m, R_inner: .5m: This scenario is like a sloth trying to run a marathon. The difference between the inner and outer radii is small, so the block's speed takes a hit. It lags behind, but hey, at least it managed to finish the race.
5) R_outer: .2m, R_inner: .1m: Ah, the little guys. Just like a snail competing against a sports car, this scenario with small radii can't even get close to the finish line. Slow and steady might win the race, but not in this case.
So, here's the final ranking:
1) R_outer: .6m, R_inner: .2m
2) R_outer: .4m, R_inner: .2m
3) R_outer: .4m, R_inner: .3m
4) R_outer: .6m, R_inner: .5m
5) R_outer: .2m, R_inner: .1m
Hope that clears things up, and remember, in the world of pulleys, the race for speed can be quite comical!
To rank these scenarios based on the speed of the block, we need to consider the concept of rotations and the relationship between the linear speed of the rope and the rotational speed of the pulley.
The linear speed of the rope is constant in all scenarios because it is being pulled at a constant speed to the right. However, the rotational speed of the pulley can vary depending on the radii.
The formula that relates linear speed (v) and rotational speed (ω) is:
v = ω * r
Where:
v = linear speed of the rope
ω = rotational speed of the pulley
r = radius of the pulley
Let's go through each scenario and assess the speed of the block:
1. R_outer: 0.4m, R_inner: 0.2m
In this scenario, the outer pulley radius is twice the inner pulley radius. Since the rope is pulled at a constant speed, the linear speed of the rope remains constant. However, the outer pulley has a larger radius, so its rotational speed is slower compared to the inner pulley. As a result, the block will have a slower speed. Rank: Slowest.
2. R_outer: 0.6m, R_inner: 0.5m
Here, the difference between the outer and inner radii is smaller. The outer pulley has a larger radius, so its rotational speed will be slower. However, the difference in radii is not as significant as in the previous scenario. Therefore, the block's speed will be faster compared to scenario 1, but still slower compared to others. Rank: Slower.
3. R_outer: 0.4m, R_inner: 0.3m
In this case, the difference between the outer and inner radii is the same as in scenario 1. Therefore, the relative rotational speed of the pulleys will be similar. The block's speed will also be slower but faster than the first two scenarios. Rank: Slower.
4. R_outer: 0.6m, R_inner: 0.2m
In this scenario, the difference in radii is greater than in scenario 2. The outer pulley has a larger radius, resulting in a slower rotational speed compared to the inner pulley. However, the difference in radii is significant, causing a faster speed for the block compared to the previous scenarios. Rank: Faster.
5. R_outer: 0.2m, R_inner: 0.1m
In this case, the pulleys have the smallest radii among all scenarios. As a result, both pulleys will rotate at a faster speed. Since the linear speed of the rope remains constant, the block will have the fastest speed in this scenario. Rank: Fastest.
So, based on the rankings, the order of the scenarios from slowest to fastest speed of the block is:
1. R_outer: 0.4m, R_inner: 0.2m
2. R_outer: 0.6m, R_inner: 0.5m
3. R_outer: 0.4m, R_inner: 0.3m
4. R_outer: 0.6m, R_inner: 0.2m
5. R_outer: 0.2m, R_inner: 0.1m
(largest)
R_outer: .6m
R_inner: .5m
R_outer: .4m
R_inner: .3m
R_outer: .4m, R_outer: .2m
R_inner: .2m , R_inner: .1m
R_outer: .6m
R_inner: .2m
(smallest)