A 9.5-kg watermelon and a 7.2-kg pumpkin are attached to each other via a cord that wraps over a pulley, as shown. Friction is negligible everywhere in this system.
(a) Find the accelerations of the pumpkin and the watermelon. Specify magnitude and direction.
magnitude direction
Pumpkin?m/s^2
Watemelon? m/s^2
To find the acceleration of the pumpkin and the watermelon, we first need to identify the forces acting on each object.
Let's assume that the pumpkin is on the left side and the watermelon is on the right side of the pulley.
For the pumpkin:
1. There is a gravitational force acting downwards with a magnitude of m₁g, where m₁ is the mass of the pumpkin (7.2 kg) and g is the acceleration due to gravity (9.8 m/s²).
2. The tension in the cord is acting upwards.
For the watermelon:
1. There is a gravitational force acting downwards with a magnitude of m₂g, where m₂ is the mass of the watermelon (9.5 kg) and g is the acceleration due to gravity (9.8 m/s²).
2. The tension in the cord is acting downwards.
Since the cord is inextensible and has negligible mass, the tension in the cord will be constant throughout its length. Therefore, the magnitude of the tension will be the same for both the pumpkin and the watermelon.
Now, let's consider the direction of the acceleration:
1. If the pumpkin has a greater acceleration than the watermelon, it means that the total force acting on the pumpkin is greater than the force acting on the watermelon. This would cause the pumpkin to move downwards and the watermelon to move upwards.
2. If the watermelon has a greater acceleration than the pumpkin, it means that the total force acting on the watermelon is greater than the force acting on the pumpkin. This would cause the watermelon to move downwards and the pumpkin to move upwards.
3. If the accelerations are equal, it means that the forces acting on both objects are equal, resulting in no net movement.
To find the magnitudes of the accelerations, we can use Newton's second law of motion:
For the pumpkin:
m₁a = m₁g - T
where T is the tension in the cord.
For the watermelon:
m₂a = T - m₂g
Since the tension is the same for both objects, we can set the two equations equal to each other:
m₁g - T = T - m₂g
2T = (m₁ + m₂)g
T = (m₁ + m₂)g / 2
Now, we can substitute the value of T back into the equations to find the accelerations:
For the pumpkin:
m₁a = m₁g - (m₁ + m₂)g / 2
a = (m₁g - (m₁ + m₂)g / 2) / m₁
a = g (m₁ - (m₁ + m₂) / 2) / m₁
For the watermelon:
m₂a = (m₁ + m₂)g / 2 - m₂g
a = ((m₁ + m₂)g / 2 - m₂g) / m₂
a = g ((m₁ + m₂) / 2 - m₂) / m₂
Substituting the given values:
m₁ = 7.2 kg
m₂ = 9.5 kg
g = 9.8 m/s²
Calculating the magnitudes and directions:
For the pumpkin:
a = 9.8 (7.2 - (7.2 + 9.5) / 2) / 7.2
a ≈ -4.74 m/s² (negative sign indicates downwards)
For the watermelon:
a = 9.8 ((7.2 + 9.5) / 2 - 9.5) / 9.5
a ≈ 4.74 m/s² (positive sign indicates upwards)
Therefore, the magnitudes and directions of the accelerations are:
Pumpkin: 4.74 m/s² downwards
Watermelon: 4.74 m/s² upwards