A tennis ball attached to the end of a light string hangs inside a car traveling on a circular track at 4.7 m/s. What is the radius of the circular track if the string makes a 10° angle with the vertical direction?
is the track vertical or horizontal?
To find the radius of the circular track, we can use the concept of centripetal acceleration.
The centripetal acceleration of an object moving in a circle is given by the equation:
a = v² / r
where "a" is the centripetal acceleration, "v" is the velocity of the object, and "r" is the radius of the circle.
In this case, we are given the velocity of the car (4.7 m/s), but to find the centripetal acceleration, we need to find the velocity of the tennis ball.
To find the velocity of the tennis ball, we can use trigonometry.
The vertical component of the velocity of the tennis ball is v * sin(theta), where theta is the angle the string makes with the vertical direction (10°).
v(vertical) = v * sin(theta)
= 4.7 m/s * sin(10°)
Next, we can use the Pythagorean theorem to find the total velocity of the tennis ball.
The horizontal component of the velocity of the tennis ball is v * cos(theta), where theta is the angle the string makes with the vertical direction (10°).
v(horizontal) = v * cos(theta)
= 4.7 m/s * cos(10°)
The total velocity of the tennis ball can be found using the Pythagorean theorem:
v = sqrt(v(vertical)² + v(horizontal)²)
= sqrt((4.7 m/s * sin(10°))² + (4.7 m/s * cos(10°))²)
Now that we have the velocity of the tennis ball, we can use the formula for centripetal acceleration to find the radius of the circle.
a = v² / r
Solving for "r", we get:
r = v² / a
Substituting the values we have:
r = (sqrt((4.7 m/s * sin(10°))² + (4.7 m/s * cos(10°))²))² / a
Since the centripetal acceleration is given by "a = g * sin(theta)" (where g is the acceleration due to gravity and theta is the angle the string makes with the vertical direction), we can substitute this into the equation:
r = (sqrt((4.7 m/s * sin(10°))² + (4.7 m/s * cos(10°))²))² / (g * sin(theta))
Finally, substituting the values of "g" (approximately 9.8 m/s²) and theta (10°), we can calculate the radius:
r = (sqrt((4.7 m/s * sin(10°))² + (4.7 m/s * cos(10°))²))² / (9.8 m/s² * sin(10°))
Using a calculator, we can evaluate this expression to find the radius of the circular track.