A 500 {\rm g} ball swings in a vertical circle at the end of a 1.5-m-long string. When the ball is at the bottom of the circle, the tension in the string is 14 {\rm N}.What is the speed of the ball at that point?
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To find the speed of the ball at the bottom of the circle, we can use the concept of centripetal force. The tension in the string provides the centripetal force that keeps the ball moving in a circle.
Step 1: Determine the mass of the ball in kilograms.
Given: Mass of the ball = 500 g = 0.5 kg
Step 2: Determine the tension in the string at the bottom of the circle.
Given: Tension in the string = 14 N
Step 3: Calculate the gravitational force acting on the ball.
Gravitational force = mass × acceleration due to gravity
Gravitational force = 0.5 kg × 9.8 m/s^2 = 4.9 N
Step 4: Calculate the net force acting on the ball at the bottom of the circle.
Net force = tension - gravitational force
Net force = 14 N - 4.9 N = 9.1 N
Step 5: Calculate the speed of the ball at the bottom of the circle using the net force.
Net force = (mass × velocity^2) / radius
9.1 N = (0.5 kg × velocity^2) / 1.5 m
Simplifying, we get: velocity^2 = (9.1 N × 1.5 m) / 0.5 kg
Velocity^2 = 27.3 m^2/s^2
Step 6: Take the square root of both sides to find the velocity.
Velocity = √(27.3 m^2/s^2)
Velocity = 5.22 m/s
Therefore, the speed of the ball at the bottom of the circle is approximately 5.22 m/s.
To find the speed of the ball at the bottom of the circle, we can use the concept of mechanical energy conservation:
1. The total mechanical energy of the ball is conserved throughout the motion, which means the sum of its kinetic energy (KE) and potential energy (PE) remains constant.
(KE + PE)₁ = (KE + PE)₂
2. At the top point, all the ball's energy is in the form of potential energy (PE). At the bottom point, all the energy is in the form of kinetic energy (KE).
PE = mgh (potential energy formula)
KE = (1/2)mv² (kinetic energy formula)
3. The tension in the string at the bottom of the circle provides the centripetal force, which is responsible for keeping the ball in circular motion.
Tension (T) = mv²/r (centripetal force formula)
Now, let's calculate:
Given data:
Mass of the ball (m) = 500 g = 0.5 kg
Length of the string (r) = 1.5 m
Tension in the string (T) = 14 N
1. Calculate the potential energy (PE) at the top:
PE = mgh
Since the ball is swinging in a vertical circle, the height (h) at the top is equal to the length of the string (r).
PE = (0.5 kg) * 9.8 m/s² * 1.5 m
PE = 7.35 J
2. Apply mechanical energy conservation:
(KE + PE)₁ = (KE + PE)₂
At the top (₁): KE₁ = 0
At the bottom (₂): PE₂ = 0
KE₂ = (KE + PE)₁ - PE₂
KE₂ = 7.35 J - 0 J
KE₂ = 7.35 J
3. Calculate the kinetic energy (KE) at the bottom:
KE = (1/2)mv²
7.35 J = (1/2) * (0.5 kg) * v²
v² = (2 * 7.35 J) / (0.5 kg)
v² = 29.4 m²/s²
4. Calculate the speed (v) at the bottom:
v = √(29.4 m²/s²)
v ≈ 5.42 m/s
Therefore, the speed of the ball at the bottom of the circle is approximately 5.42 m/s.