Question

two coins are placed on a turntable record player that is spinning at 33 rpm . coin 1 is 2 cm from the center. coin 2 is 8 cm from the center. what is the ratio of the centripetal accelerations a2/a1 for the coins?

Answer 1

The ratio of centripetal accelerations for two coins on a turntable spinning at 33 rpm, with one 2 cm from the center and the other 8 cm from the center, is 16.

Step-by-step explanation:

Calculating the Ratio of Centripetal Accelerations

Centripetal acceleration is given by the formula a = rω², where a is the centripetal acceleration, r is the radius from the center of rotation, and ω is the angular velocity in radians per second. Since both coins are on the same turntable spinning at 33 revolutions per minute (rpm), they share the same angular velocity. The conversion from rpm to radians per second is necessary to use ω in the centripetal acceleration formula.

To compare accelerations a2/a1, we only need to compare the squares of their radius ratios since ω is constant and cancels out:

ω = 33 rev/min * (2π rad/rev) * (1 min/60 s) = 3.49 rad/s

Now, let's calculate the centripetal accelerations:

• For coin 1 at r1 = 2 cm: a1 = r1ω²
• For coin 2 at r2 = 8 cm: a2 = r2ω²

The ratio of centripetal accelerations a2/a1 is then:

a2/a1 = (r2/r1)² = (8 cm / 2 cm)² = 4² = 16

Therefore, the ratio of the centripetal accelerations for the two coins is 16.

Answer 2

The ratio of the centripetal accelerations a2 to a1 for the two coins is simply the ratio of their radii, which is 4.

Step-by-step explanation:

To find the ratio of the centripetal accelerations a2 to a1 for two coins placed at different distances from the center of a spinning turntable, we can use the formula for centripetal acceleration:

ac = ω2r

where ac is the centripetal acceleration, ω is the angular velocity in radians per second, and r is the radius, or the distance from the center.

First, convert 33 revolutions per minute (rpm) to radians per second:

ω = 33 rpm × (2π rad/rev) × (1 min/60 s)

Now, we can calculate the individual centripetal accelerations:

a1 = ω2 × r1

a2 = ω2 × r2

For coin 1 (r1 = 2 cm):

a1 = ω2 × 0.02 m

For coin 2 (r2 = 8 cm):

a2 = ω2 × 0.08 m

The ratio of the centripetal accelerations is given by:

a2/a1 = (r2/r1)

Since the angular velocity (ω) is the same for both coins, it cancels out in the ratio, and we are left with a simple ratio of the radii:

a2/a1 = (0.08 m/0.02 m)

The ratio is:

a2/a1 = 4