a sledge of mass 25kg is pulled along a smooth frozen lake by a rope which is pulled at an angle of 30 degrees to the horizontal.if the sledge accelerates at 1.5 metres per second, find the tension.
To find the tension in the rope, we can start by analyzing the forces acting on the sledge. There are two main forces: the force of tension in the rope and the force of gravity.
1. Decompose the force of tension:
Since the rope is pulled at an angle of 30 degrees to the horizontal, we need to split the force of tension into its horizontal and vertical components. Let's call the tension force "T."
The vertical component (T⋅sinθ) acts opposite to the force of gravity (mg), where "m" is the mass of the sledge and "g" is the acceleration due to gravity (approximately 9.8 m/s²).
The horizontal component (T⋅cosθ) is responsible for accelerating the sledge.
2. Write the equations of motion:
For the horizontal direction: The force of tension (T⋅cosθ) is equal to the mass of the sledge (m) multiplied by the acceleration (a).
T⋅cosθ = m⋅a
For the vertical direction: The force of tension (T⋅sinθ) minus the force of gravity (mg) is equal to zero since the sledge is not moving vertically.
(T⋅sinθ) - mg = 0
3. Substitute values and solve the equations:
Given: mass (m) = 25 kg, angle (θ) = 30 degrees, and acceleration (a) = 1.5 m/s².
For the horizontal direction:
T⋅cos30 = 25⋅1.5
T⋅√3/2 = 37.5
T = 37.5⋅2/√3
T ≈ 43.3 N (rounded to one decimal place)
Therefore, the tension in the rope is approximately 43.3 Newtons.