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?
To solve this problem, we can use the relative size of the radii to find the relationship between the angles turned by the larger and smaller pulleys.
Recall that the circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius.
For the smaller pulley with a radius of 2 inches, the circumference is C1 = 2π(2) = 4π inches.
Since the smaller pulley turns through 50 radians, we can set up a proportion to relate the angles turned by the pulleys:
Angle turned by larger pulley / Angle turned by smaller pulley = Circumference of larger pulley / Circumference of smaller pulley
Let x be the angle turned by the larger pulley. We then have:
x / 50 radians = 2π(5) / 4π
Simplifying the right side gives:
x / 50 radians = 5 / 2
Cross-multiplying gives:
2x = 50 * 5
2x = 250
x = 250 / 2
x = 125 radians
Therefore, the larger pulley will turn through 125 radians.