A 84 g ball is fastened to one end of a string
52 cm long and the other end is held fixed at
point O so that the string makes an angle of
23� with the vertical, as in the figure. This
angle remains constant as the ball rotates in
a horizontal circle. The angle � would remain
constant only for a particular speed of the ball
To find the speed of the ball which would keep the angle θ constant, we can use the concept of centripetal force.
1. First, let's analyze the forces acting on the ball.
- The weight of the ball acts vertically downward with a magnitude of mg, where m is the mass of the ball and g is the acceleration due to gravity.
- The tension in the string acts horizontally towards the center of the circle.
- The vertical component of tension cancels out with the weight of the ball, so the only force in the vertical direction is the vertical component of tension (T * cos θ).
- The horizontal component of tension (T * sin θ) provides the centripetal force needed to keep the ball moving in a circle.
2. The centripetal force required to maintain circular motion is given by the formula:
Fc = m * v^2 / r
where Fc is the centripetal force, m is the mass of the ball, v is the speed of the ball, and r is the radius of the circle.
3. In this case, the radius of the circle is given by the length of the string, which is 52 cm or 0.52 m.
4. The horizontal component of tension provides the centripetal force:
T * sin θ = Fc
T * sin θ = m * v^2 / r
5. Rearranging the equation, we get:
T = (m * v^2 / r) / sin θ
6. Since we are looking for the speed of the ball (v), we need an expression for tension (T). We can find T using the vertical component of tension and the weight of the ball.
T * cos θ = m * g
7. Solving for T, we get:
T = m * g / cos θ
8. Substituting the expression for T in terms of m, g, and θ into the equation for T in terms of m, v, r, and θ, we have:
m * g / cos θ = (m * v^2 / r) / sin θ
9. Rearranging the equation to solve for v^2, we get:
v^2 = (g * r) / sin θ
10. Taking the square root of both sides, we obtain:
v = √((g * r) / sin θ)
Using the given values:
- Mass of the ball (m) = 84 g = 0.084 kg
- Length of the string (r) = 52 cm = 0.52 m
- Angle with the vertical (θ) = 23º = 23 * (π/180) radians (converting to radians)
- Acceleration due to gravity (g) = 9.8 m/s^2 (approximate value)
We can now substitute these values into our equation to calculate the speed of the ball (v).
To determine the specific speed at which the angle remains constant, we need to consider the forces acting on the ball.
In this case, the ball is rotating in a horizontal circle, so there must be a centripetal force acting towards the center of the circle, provided by the tension in the string.
At the same time, there is the force of gravity acting vertically downwards. The vertical component of the tension force must balance the force of gravity to keep the ball in equilibrium.
To find the speed at which the angle remains constant, we can set up equations for the forces involved.
1. Calculate the tension in the string:
The vertical component of the tension force can be found using trigonometry:
Tension (vertical component) = T * cos(23°)
2. Calculate the force of gravity:
The force of gravity acting on the ball is given by:
Force of gravity = mass * acceleration due to gravity = 84 g * 9.8 m/s^2
3. Set up an equation to find the speed:
Since the angle remains constant, the vertical component of the tension force must balance the force of gravity. Therefore,
T * cos(23°) = 84 g * 9.8 m/s^2
To find the speed, we need to relate the tension in the string to the speed of the ball. The tension in the string is also related to the centripetal force as:
Tension = m * v^2 / r
where m is the mass of the ball, v is the speed of the ball, and r is the radius of the circular motion.
4. Calculate the radius:
The radius of the circular motion can be found using trigonometry:
radius = string length * sin(23°)
5. Plug in the values and solve for the speed:
Plugging in the values from the problem, we can solve the equation:
m * v^2 / r * cos (23°) = 84 g * 9.8 m/s^2
Solve this equation for v to find the specific speed at which the angle remains constant.