A bowling ball weighing 71.2 N is attached to the ceiling by a 3.60 m rope. The ball is pulled to one side and released; it then swings back and forth like a pendulum. As the rope swings through its lowest point, the speed of the bowling ball is measured at 4.00 m/s. At that instant, find the magnitude of the acceleration of the bowling ball. At that instant, find the tension in the rope.
mass of ball = 71.2/9.81 = 7.26 kg
Acentripetal = v^2/R = 16/3.6 up
Force in rope = T up
force of gravity down = m g = 71.2 N
acceleration up = 16/3.6
F = m a
T -71.2 = 16/3.6
T = 71.2 + 16/3.6
Student itt tech
To find the magnitude of the acceleration of the bowling ball at its lowest point, we can use the equation for calculating the acceleration of an object in simple harmonic motion:
a = (v^2) / r
Where:
a = acceleration
v = velocity
r = distance from the center of motion (radius)
In this case, the velocity of the bowling ball at its lowest point is given as 4.00 m/s, and the distance from the center of motion is equal to the length of the rope, which is 3.60 m.
Substituting the given values into the equation, we have:
a = (4.00^2) / 3.60
a = 16.00 / 3.60
a ≈ 4.44 m/s^2
Therefore, the magnitude of the acceleration of the bowling ball at its lowest point is approximately 4.44 m/s^2.
To find the tension in the rope at that instant, we can consider the forces acting on the bowling ball. At its lowest point, the tension in the rope provides the centripetal force to keep the ball in circular motion.
The centripetal force is given by the equation:
Fc = m * a
Where:
Fc = centripetal force
m = mass
a = acceleration
In this case, the weight of the bowling ball is given as 71.2 N, which is equal to the force of gravity acting on it. We can convert this force to mass using Newton's second law:
Fg = m * g
Where:
Fg = force of gravity
m = mass
g = acceleration due to gravity (approximately 9.8 m/s^2)
Solving for mass:
m = Fg / g
m = 71.2 N / 9.8 m/s^2
m ≈ 7.27 kg
Substituting the mass and acceleration values into the equation for centripetal force, we have:
Fc = 7.27 kg * 4.44 m/s^2
Fc ≈ 32.37 N
Therefore, the tension in the rope at that instant is approximately 32.37 N.
To find the magnitude of the acceleration of the bowling ball at the lowest point, we can use the concept of centripetal acceleration. When an object moves in a circular path, it experiences centripetal acceleration toward the center of the circle.
First, let's calculate the length of the rope, which is also the distance traveled by the bowling ball. We have a 3.60 m rope, so the total distance traveled is 2 times this value (since the ball swings back and forth): 2 * 3.60 m = 7.20 m.
Now, we can use the formula for centripetal acceleration:
a = v^2 / r
where a is the acceleration, v is the speed of the ball at the lowest point, and r is the radius of the circular motion.
In this case, since the ball is attached to the ceiling and moves in a circular path, the radius of the path is equal to the length of the rope: r = 3.60 m.
Plugging in the given values, we get:
a = (4.00 m/s)^2 / 3.60 m
a = 16.00 m^2/s^2 / 3.60 m
a ≈ 4.44 m/s^2 (rounded to two decimal places)
So, the magnitude of the acceleration of the bowling ball at the lowest point is approximately 4.44 m/s^2.
Next, let's find the tension in the rope at that instant. At the lowest point of the swing, two forces are acting on the ball: its weight (mg) and the tension in the rope (T). The net force is the centripetal force that provides acceleration towards the center of the circular path.
We can use Newton's second law of motion to find the tension:
ΣF = ma
Since the ball is moving in a circular path at a constant speed, the net force acting on it is the centripetal force.
ΣF = Fc = m * a
The centripetal force is provided by the tension in the rope:
Fc = T
Therefore, we have:
m * a = T
Plugging in the given values:
m = 71.2 N (weight)
a = 4.44 m/s^2 (acceleration)
T = 71.2 N * 4.44 m/s^2
T ≈ 316.61 N (rounded to two decimal places)
So, the tension in the rope at the instant when the speed of the bowling ball is measured at 4.00 m/s is approximately 316.61 N.