A bowling ball weighing 71.8 N is attached to the ceiling by a rope of length 3.83 m. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 4.50 m/s.
What is the acceleration of the bowling ball, in magnitude and direction, at this instant?
What is the tension in the rope at this instant?
First, we can find the mass of the bowling ball using its weight:
m = 71.8 N / 9.81 m/s^2
m = 7.32 kg
Next, let's find the centripetal acceleration of the bowling ball as it swings through the vertical.
ac = v^2 / r
ac = (4.50 m/s)^2 / 3.83 m
ac = 5.31 m/s^2
The centripetal acceleration is directed towards the center of the circle, so at the instant when the rope is vertical, the centripetal acceleration is upwards.
Next, we can find the tension in the rope at this instant. The tension must provide two forces: one to balance the weight of the bowling ball and the other to provide the centripetal acceleration. So, the tension can be calculated as:
T = m * (g + ac)
T = 7.32 kg * (9.81 m/s^2 + 5.31 m/s^2)
T = 7.32 kg * 15.12 m/s^2
T = 110.68 N
So at that instant, the acceleration of the bowling ball is 5.31 m/s^2 upwards, and the tension in the rope is 110.68 N.
To find the acceleration of the bowling ball at the instant when it swings through the vertical, we can apply the equations of motion for a simple pendulum.
The acceleration of a simple pendulum can be given by the formula:
a = (g * θ) / L
Where:
a = acceleration of the pendulum
g = acceleration due to gravity (approximately 9.8 m/s²)
θ = angular displacement (in radians) of the pendulum from the vertical position
L = length of the pendulum
In this case, we are given the speed of the ball, but we need to find the angular displacement. The ball reaches its maximum speed when it passes through the equilibrium position (the vertical position in this case). At this point, its kinetic energy is at its maximum and its potential energy is at its minimum. Therefore, the ball must be at its maximum angular displacement.
We can use the equation for the kinetic energy of the pendulum at its maximum displacement to find the angular displacement (θ):
KE = (1/2) * m * v² = m * g * L * (1 - cos(θ))
Where:
KE = kinetic energy
m = mass of the ball
v = speed of the ball
g = acceleration due to gravity
L = length of the pendulum
θ = angular displacement (to be found)
Rearranging the equation, we have:
θ = cos^(-1)((m * g * L - (1/2) * m * v²) / (m * g * L))
Now, we can calculate the value of θ using the given values:
m = 71.8 N / 9.8 m/s² = 7.33 kg
v = 4.50 m/s
L = 3.83 m
g = 9.8 m/s²
Substituting these values into the equation, we find:
θ = cos^(-1)((7.33 kg * 9.8 m/s² * 3.83 m - (1/2) * 7.33 kg * (4.50 m/s)²) / (7.33 kg * 9.8 m/s² * 3.83 m))
After calculating θ, we can substitute it into the equation for acceleration:
a = (g * θ) / L
Again, using the given values, we can find the value of acceleration (a).
For the tension in the rope, at this instant, it is equal to the gravitational force acting on the ball plus the centripetal force required to keep the ball moving in a circular path.
The equation for tension (T) in the rope can be given by:
T = m * g + (m * v²) / L
Substituting the given values, we can calculate the tension in the rope (T).
Please note that the above explanation assumes ideal conditions and neglects any other factors that may affect the accuracy of the results, such as air resistance and friction.