A tether ball of mass 0.25 kg is attached to a vertical pole by a cord 1.4 m long. Assume the cord attaches to the center of the ball. The cord makes an angle of 20° with the vertical.
(a) What is the tension in the cord?
(b) What is the speed of the ball?
r = radius of circle = 1.4 sin 20
centripetal acceleration = v^2/r
horizontal force needed = m v^2/r
but
horizontal component of tension = T sin 20
vertical component of tension = T cos 20 = m g
so
T = m g/cos 20 = 2.61 Newtons (part A)
so
T sin 20 = m v^2/r = m v^2/1.4 sin 20
2.61 (1.4 )sin^2 20 = .25 v^2
solve for v
(a) Well, you know what they say, tension is a lot like that feeling when you try to open a bag of chips, but it just won't budge. In this case, we're dealing with a tether ball, not a bag of chips, but the concept is similar. We can find the tension in the cord by using some good old trigonometry.
The tension in the cord can be found using the equation T = mgcosθ, where T is the tension, m is the mass of the ball, g is the acceleration due to gravity, and θ is the angle the cord makes with the vertical.
Plugging in the values, we get T = (0.25 kg)(9.8 m/s^2)cos(20°). Crunching the numbers, we find that the tension in the cord is approximately 2.17 N. Ta-da!
(b) Now, let's move on to the speed of the ball. We can use some physics equations to tackle this one. The speed of an object in circular motion can be calculated using the equation v = ωr, where v is the speed, ω is the angular velocity, and r is the radius of the circular path.
In this case, the ball is swinging around the pole in a circular motion, so the radius of the circular path is the length of the cord, which is 1.4 m.
To find the angular velocity, we can use the equation ω = √(g/r), where g is the acceleration due to gravity and r is the radius of the circular path.
Plugging in the values, we get ω = √(9.8 m/s^2 / 1.4 m) ≈ 2.74 rad/s.
Finally, we can find the speed of the ball by multiplying the angular velocity by the radius: v = (2.74 rad/s)(1.4 m) ≈ 3.84 m/s.
So, the speed of the ball is approximately 3.84 m/s. Whoosh!
To find the tension in the cord, we can break down the forces acting on the tetherball.
(a) The force of tension in the cord can be found using the equation:
Tension = Weight of the ball + Centripetal force
The weight of the ball can be calculated using:
Weight = mass * gravity
Where,
mass = 0.25 kg (given)
gravity = 9.8 m/s^2 (acceleration due to gravity)
Weight = 0.25 kg * 9.8 m/s^2 = 2.45 N
The centripetal force can be calculated using:
Centripetal force = (mass * speed^2) / radius
Here, the radius is given as 1.4 m.
Since the cord attaches to the center of the ball, the radius is equal to the length of the cord.
Centripetal force = (0.25 kg * speed^2) / 1.4 m
Now, we can set up an equation for the tension:
Tension = Weight + Centripetal force
Substituting the values we found:
Tension = 2.45 N + (0.25 kg * speed^2) / 1.4 m
(b) To find the speed of the ball, we can use the centripetal force equation mentioned above:
Centripetal force = (0.25 kg * speed^2) / 1.4 m
We can rearrange this equation to solve for the speed:
Speed = √((Centripetal force * 1.4 m) / 0.25 kg)
Now, we can substitute the centripetal force calculated in part (a) into this equation to find the speed.
To find the tension in the cord, we can consider the forces acting on the tether ball.
(a) The tension in the cord can be found by analyzing the forces acting on the ball. The two forces acting on the ball are its weight (mg) and the tension in the cord (T). Since the ball is in equilibrium, meaning it is not accelerating, the sum of the forces in the vertical direction must be zero.
The weight of the ball can be calculated by multiplying its mass (m) by the acceleration due to gravity (g). In this case, m = 0.25 kg and g ≈ 9.8 m/s^2.
The vertical component of the tension (T_y) can be found by multiplying the tension (T) by the sine of the angle the cord makes with the vertical (θ). In this case, θ = 20°.
Setting up an equation for the forces in the vertical direction:
T_y - mg = 0
Substituting the values:
T * sin(20°) - (0.25 kg * 9.8 m/s^2) = 0
Simplifying the equation:
T * sin(20°) = 0.25 kg * 9.8 m/s^2
Finally, solving for T, the tension in the cord:
T = (0.25 kg * 9.8 m/s^2) / sin(20°)
Calculating the value:
T ≈ 1.44 N (to two significant figures)
Therefore, the tension in the cord is approximately 1.44 N.
(b) To find the speed of the ball, we can use the concept of centripetal force.
The centripetal force acting on the ball is provided by the tension in the cord. The centripetal force (F_c) can be calculated by multiplying the mass (m) of the ball by the square of its velocity (v) and dividing it by the radius (r) of the circular path.
In this case, the radius (r) is equal to the length of the cord, which is given as 1.4 m. The mass (m) and the tension (T) have been given as 0.25 kg and 1.44 N, respectively.
Therefore, setting up an equation:
F_c = T = (m * v^2) / r
Rearranging the equation to solve for v, the speed of the ball:
v = sqrt((T * r) / m)
Substituting the values:
v = sqrt((1.44 N * 1.4 m) / 0.25 kg)
Calculating the value:
v ≈ 5.29 m/s (to three significant figures)
Therefore, the speed of the ball is approximately 5.29 m/s.