A tennis ball connected to a string is spun around in a vertical, circular path at a uniform speed. The ball has a mass m = 0.167 kg and moves at v = 4.9 m/s. The circular path has a radius of R = 0.98 m
1.)What is the magnitude of the tension in the string when the ball is at the bottom of the circle?
2.)What is the magnitude of the tension in the string when the ball is at the side of the circle?
3.)What is the magnitude of the tension in the string when the ball is at the top of the circle?
the centripetal force (tension in the string)
... keeps the ball moving in a circle
the force necessary is ... m v^2 / r = .167 * 4.9^2 / .98 ... Newtons
... this is the tension at the sides of the circle
at the top and bottom of the circle
... the weight of the ball adds to or subtracts from the string tension
Where are the equations, worked with equations?
To answer these questions, we will use the concepts of centripetal force and gravitational force.
1.) Tension at the bottom of the circle:
At the bottom of the circle, the tension in the string will be the sum of the centripetal force and the gravitational force acting on the ball.
The centripetal force(Fc) is given by the equation Fc = mv^2/R, where m is the mass of the ball, v is its velocity, and R is the radius of the circular path.
The gravitational force(Fg) acting on the ball is given by the equation Fg = mg, where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, the tension (T) in the string at the bottom of the circle will be T = Fc + Fg.
2.) Tension at the side of the circle:
At the side of the circle, the tension in the string will only be the centripetal force acting on the ball because the gravitational force is acting perpendicular to the circular motion and does not contribute to the tension in the string.
Therefore, the tension (T) in the string at the side of the circle will be T = Fc.
3.) Tension at the top of the circle:
At the top of the circle, the tension in the string will again be the sum of the centripetal force and the gravitational force acting on the ball.
So, the tension (T) in the string at the top of the circle will be T = Fc + Fg.
Let's calculate the values for each question:
1.) Tension at the bottom of the circle:
Fc = mv^2/R
= (0.167 kg)(4.9 m/s)^2 / 0.98 m
= 0.0835 kg m/s^2
Fg = mg
= (0.167 kg)(9.8 m/s^2)
= 1.6386 N
T = Fc + Fg
= 0.0835 kg m/s^2 + 1.6386 N
= 1.7221 N
Therefore, the magnitude of the tension in the string when the ball is at the bottom of the circle is 1.7221 N.
2.) Tension at the side of the circle:
Fc = mv^2/R
= (0.167 kg)(4.9 m/s)^2 / 0.98 m
= 0.0835 kg m/s^2
Therefore, the magnitude of the tension in the string when the ball is at the side of the circle is 0.0835 kg m/s^2 (same as the bottom).
3.) Tension at the top of the circle:
Fc = mv^2/R
= (0.167 kg)(4.9 m/s)^2 / 0.98 m
= 0.0835 kg m/s^2
Fg = mg
= (0.167 kg)(9.8 m/s^2)
= 1.6386 N
T = Fc + Fg
= 0.0835 kg m/s^2 + 1.6386 N
= 1.7221 N
Therefore, the magnitude of the tension in the string when the ball is at the top of the circle is 1.7221 N (same as the bottom).
To find the tension in the string at different points in the circular path, we need to consider the forces acting on the object at each point. The ball is undergoing uniform circular motion, which means there must be a net force acting towards the center of the circle to keep it moving in a circular path.
1.) When the ball is at the bottom of the circle:
At the bottom of the circle, the weight of the ball acts downwards, while the tension in the string acts upwards. The tension in the string provides the necessary centripetal force to keep the ball moving in a circular path. To find the tension, we can equate the centripetal force to the net force acting on the object.
Net force = Centripetal force
Tension + Weight = (mass * velocity^2) / radius
The weight of the object is given by the formula:
Weight = mass * acceleration due to gravity
First, calculate the weight of the object:
Weight = 0.167 kg * 9.8 m/s^2
Next, calculate the centripetal force:
Centripetal force = (0.167 kg * (4.9 m/s)^2) / 0.98 m
Finally, find the tension by subtracting the weight from the centripetal force:
Tension = Centripetal force - Weight
2.) When the ball is at the side of the circle:
At the side of the circle, the weight of the ball still acts downwards, while the tension in the string acts towards the center of the circle. The tension still provides the necessary centripetal force. The net force acting on the object is still the centripetal force, so the tension will be equal to the centripetal force.
Centripetal force = (mass * velocity^2) / radius
Calculate the centripetal force using the given values.
3.) When the ball is at the top of the circle:
At the top of the circle, the weight of the ball still acts downwards, while the tension in the string acts downwards as well (opposite to the direction of motion). The net force acting on the object is the difference between the weight and the centripetal force.
Centripetal force = (mass * velocity^2) / radius
Calculate the centripetal force and the weight. Subtract the centripetal force from the weight to get the magnitude of the tension.