A 30.0 ball rolls around a 50.0 -diameter L-shaped track, shown in the figure, at 60.0 . What is the magnitude of the net force that the track exerts on the ball? Rolling friction can be neglected.
To find the magnitude of the net force that the track exerts on the ball, we can start by considering the forces acting on the ball.
1. Gravity: The ball experiences a downward force due to gravity. The magnitude of this force can be calculated using the formula: F_gravity = m * g, where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s²).
2. Normal force: The ball is in contact with the track, and the track exerts an upward normal force on the ball to support its weight. The magnitude of this force is equal to the weight of the ball, which is given by: F_normal = m * g.
3. Centripetal force: Since the ball is rolling around the track, there must be a centripetal force acting towards the center of the circular path. This force is provided by the track. The magnitude of the centripetal force can be calculated using the formula: F_centripetal = (m * v²) / r, where v is the velocity of the ball and r is the radius of the track (half the diameter).
Now, let's calculate the magnitudes of these forces and find the net force:
1. Gravity: Given the mass of the ball (m = 30.0 kg), we can calculate the force due to gravity: F_gravity = (30.0 kg) * (9.8 m/s²) = 294 N.
2. Normal force: Since the ball is not accelerating vertically, the normal force is equal to the weight of the ball: F_normal = 294 N.
3. Centripetal force: The velocity of the ball is 60.0 m/s. The radius of the track is half of the diameter, so r = 50.0 / 2 = 25.0 m. Calculating the centripetal force: F_centripetal = (30.0 kg * (60.0 m/s)²) / 25.0 m = 4320 N.
The net force is the vector sum of these forces. Since the normal force and centripetal force are in the same direction, we can add them together: net force = F_normal + F_centripetal = 294 N + 4320 N = 4614 N.
Therefore, the magnitude of the net force that the track exerts on the ball is 4614 N.
In order to determine the magnitude of the net force that the track exerts on the ball, we need to consider the centripetal force acting on the ball as it moves around the track.
The centripetal force is given by the equation:
Fc = mv² / r
Where:
- Fc is the centripetal force
- m is the mass of the ball
- v is the velocity of the ball
- r is the radius of the circular path or track
Given:
- Mass of the ball (m) = 30.0 g = 0.030 kg
- Diameter of the track (d) = 50.0 cm = 0.50 m
- Velocity of the ball (v) = 60.0 m/s
First, we need to find the radius of the track, which is half of the diameter:
r = d / 2 = 0.50 m / 2 = 0.25 m
Now, we can calculate the centripetal force:
Fc = mv² / r
Fc = (0.030 kg) × (60.0 m/s)² / 0.25 m
Calculating:
Fc = (0.030 kg) × (3600 m²/s²) / 0.25 m
Fc = (0.030 kg) × 14400 m²/s² / 0.25 m
Fc = 432 kg·m/s² = 432 N
Therefore, the magnitude of the net force that the track exerts on the ball is equal to 432 Newtons.