A ball is fastened to one en of a 28 cm string, and the other end is held fixed to a support. The ball whirls in a horizontal circle. Find the speed of the ball in its circular path if the string makes an angle of 35 degrees to the vertical.
A free body diagram of forces acting on the ball should tell you that
Tan 35 = (mg)/(mv^2/R)
= R g/ v^2
The mass m cancels out.
Solve for v.
To find the speed of the ball in its circular path, we can use the concept of centripetal force. Centripetal force is the force that keeps an object moving in a circular path. In this case, the tension in the string provides the centripetal force.
The tension in the string can be split into two components: one vertical and one horizontal. The vertical component is equal to the weight of the ball, and the horizontal component provides the centripetal force.
First, let's find the tension in the string. We can use the vertical component of the tension equation:
Tension_vertical = Weight_of_ball
The weight of the ball can be calculated using the formula:
Weight_of_ball = mass * gravity
Given that the string makes an angle of 35 degrees to the vertical, the vertical component of the tension can be represented as:
Tension_vertical = Tension * cos(angle)
Now, let's find the horizontal component of the tension. Since this component provides the centripetal force, we can equate it to:
Tension_horizontal = (mass * velocity^2) / radius
Where:
- mass is the mass of the ball
- velocity is the speed of the ball
- radius is the length of the string
Now, equating the vertical and horizontal components of the tension, we have:
Tension * cos(angle) = (mass * velocity^2) / radius
Solving for velocity, we get:
velocity = sqrt((Tension * cos(angle) * radius) / mass)
Substituting the given values:
- Tension = Tension_horizontal = (mass * velocity^2) / radius
- angle = 35 degrees
- radius = 28 cm = 0.28 m (converting to meters)
We can rewrite the equation as:
velocity = sqrt((Tension * cos(35 degrees) * 0.28 m) / mass)
Now, we need to find the value of tension. To do this, we can use the known parameters of the problem.
The vertical component of the tension can be found using the equation:
Tension_vertical = Tension * sin(angle)
Given that the tension in the vertical direction is equal to the weight of the ball:
Tension * sin(35 degrees) = weight
Substituting the weight equation:
Tension * sin(35 degrees) = mass * gravity
We can solve for tension:
Tension = (mass * gravity) / sin(35 degrees)
Substituting the known values:
- mass
- gravity = 9.8 m/s^2
Now we can substitute the tension value back into the velocity equation to get the final solution.
By following these steps and using the given information, you can find the speed of the ball in its circular path.