To analyze the system of two objects connected by a light string over a frictionless pulley and inclined plane, we can proceed as follows:
Free Body Diagram (FBD)
-
Object m1 (2 kg):
- Acts downward: Weight (W1) = \( m_1 \cdot g = 2 \text{ kg} \cdot 9.81 \text{ m/s}^2 = 19.62 \text{ N} \)
- Tension (T) acting upward.
Forces on m1:
- \( W_1 = 19.62 \text{ N} \)
- \( T \)
-
Object m2 (6 kg):
- Acts downward: Weight (W2) = \( m_2 \cdot g = 6 \text{ kg} \cdot 9.81 \text{ m/s}^2 = 58.86 \text{ N} \)
- Due to the incline, we can resolve the weight of m2 into two components:
- Perpendicular to the incline: \( W_{2\perp} = W_2 \cdot \cos(55^\circ) \)
- Parallel to the incline: \( W_{2\parallel} = W_2 \cdot \sin(55^\circ) \)
- Tension (T) acting up the incline.
Forces on m2:
- \( W_2 = 58.86 \text{ N} \)
- \( T \) acting up the incline
- \( W_{2\parallel} = 58.86 \cdot \sin(55^\circ) \)
- \( W_{2\perp} = 58.86 \cdot \cos(55^\circ) \)
Equations of Motion
For both masses, we can set up the equations of motion using Newton's second law.
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For m1 (2 kg): \[ m_1 g - T = m_1 a \] \[ 19.62 - T = 2a \quad (1) \]
-
For m2 (6 kg): The net force down the incline is given by: \[ m_2 g \sin(\theta) - T = m_2 a \] \[ 58.86 \sin(55^\circ) - T = 6a \quad (2) \]
Calculate the components for m2
Calculate \( W_{2\parallel} \): \[ W_{2\parallel} = 58.86 \cdot \sin(55^\circ) \approx 58.86 \cdot 0.8192 \approx 48.21 \text{ N} \]
Substitute and Solve
Using equations (1) and (2):
From equation (1): \[ T = 19.62 - 2a \quad (3) \]
Substituting equation (3) into equation (2): \[ 48.21 - (19.62 - 2a) = 6a \] \[ 48.21 - 19.62 + 2a = 6a \] \[ 28.59 = 4a \] \[ a = \frac{28.59}{4} \approx 7.15 \text{ m/s}^2 \]
Speed After 2 Seconds
Since both masses start from rest, we can use the kinematic equation to find the speed (\( v \)): \[ v = u + at \] Where \( u = 0 \) (initial speed), \[ v = 0 + 7.15 \cdot 2 \approx 14.3 \text{ m/s} \]
Final Results
- Acceleration (a): \( 7.15 \text{ m/s}^2 \)
- Speed after 2 seconds for both objects: \( v \approx 14.3 \text{ m/s} \)
Both objects will be moving with the same speed due to the constraints of the system.