A belt connects a pulley of 2-inch radius with a pulley of 5-inch radius. If the smaller pulley turns through 50 radians, through how many radians will the larger pulley turn?
The relationship between the radii and the angles turned by the pulleys is given by the formula:
radius_1 * angle_1 = radius_2 * angle_2
Plugging in the given values:
2 * 50 = 5 * angle_2
100 = 5 * angle_2
Dividing both sides of the equation by 5:
20 = angle_2
The larger pulley will turn through 20 radians. Answer: \boxed{20}.
To find out how many radians the larger pulley will turn, we can use the concept of rotational motion and the ratio of the radii of the pulleys.
The formula to relate the rotation of two pulleys connected by a belt is:
(r1 / r2) = (θ1 / θ2)
where r1 and r2 are the radii of the two pulleys, and θ1 and θ2 are the rotations of the two pulleys, respectively.
In this case, the smaller pulley has a radius of 2 inches and has rotated through 50 radians. The larger pulley has a radius of 5 inches and we need to find the rotation in radians, θ2.
Using the formula, we can rearrange it to solve for θ2:
r1 / r2 = θ1 / θ2
2 / 5 = 50 / θ2
Now, to isolate θ2, we can cross multiply:
2 * θ2 = 5 * 50
2 * θ2 = 250
Now, divide both sides by 2 to solve for θ2:
θ2 = 250 / 2
θ2 = 125 radians
Therefore, the larger pulley will turn through 125 radians.