Ball with a rope on the incline (fixed)

Classical Mechanics Level 3

A ball with m=300kg is connected to the top of a incline with θ=53° by a rope,as the picture shown.Initially,the system is rest and the rope is parallel to the incline. Find the tension of the rope (in newtons) when the system is accelerating to the right with a constant acceleration a=10m/s^2 .

(sin 53°=0.8,cos 53°=0.6,g=10m/s^2,Ignore all the frictions)

(Hint:If the acceleration is higher than a certain value,the ball will lose contact with the incline)

The answer is 4242.6406871192851464050661726291.

1 solution

Steven Chase
Jan 18, 2018

This is an interesting problem, because there are two possible scenarios, depending on the relationships between the problem parameters:

Possibility #1: The ball remains in contact with the ramp
Possibility #2: The ball loses contact with the ramp

Let's examine the first possibility to see whether the ball remains in contact with the ramp. Define the tension as $T$ and the ramp normal force magnitude as $N$ . The horizontal and vertical balance equations are the following ( $a_r$ is the horizontal acceleration):

$T \, cos \theta - N \, sin \theta = m \, a_r \\ T \, sin \theta + N \, cos \theta = m \, g$

Plugging in numbers and solving the linear system results in a negative value for $N$ , which is physically unfeasible, because a ramp surface can only push and cannot pull. Therefore, we know that the ball loses contact with the ramp. The only forces on the ball are the tension and gravity, and the balance equations become more trivial in this case. Additionally, we are no longer concerned with the ramp angle.

$T_x = m \, a_r = 3000 \\ T_y = m \, g = 3000 \\ T = 3000 \sqrt{2} \approx \, 4242.64$

Ball with a rope on the incline (fixed)

The answer is 4242.6406871192851464050661726291.

1 solution

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