Question

A large cubical block of ice of mass \(M=64.0 \mathrm{~kg}\) and sides of length \(L=0.400 \mathrm{~m}\) is held stationary on a frictionless ramp. The ramp is at an angle of \(\theta=26.0^{\circ}\) above the horizontal. The ice cube is held in place by a rope of negligible mass and length \(l=1.60 \mathrm{~m}\). The rope is attached to the surface of the ramp and to the upper edge of the ice cube, a distance \(L\) above the surface of the ramp. Find the tension in the rope.

Short answer

Answer: The tension in the rope is approximately 278.22 Newtons.

Step-by-step solution

1. Identifying Forces

First, we need to identify all the forces acting on the ice block. There are three main forces: the gravitational force (weight of the block) acting downwards, the normal force exerted by the ramp acting perpendicular to the ramp surface, and the tension force from the rope.

2. Breaking Down the Gravitational Force

We need to break down the gravitational force \(F_g\) acting downward into components parallel and perpendicular to the ramp. To do this, we can use the angle of inclination \(\theta\): - The component parallel to the ramp: \(F_g\sin{\theta}\) - The component perpendicular to the ramp: \(F_g\cos{\theta}\) where \(F_g = Mg\)

3. Analyzing Forces in Equilibrium

Since the ice block is stationary, it is in equilibrium. Therefore, the net forces acting on the ice block along the ramp and perpendicular to the ramp must be both zero. We can write two equations of equilibrium: 1. Forces along the ramp: \(T - F_g\sin{\theta} = 0\) 2. Forces perpendicular to the ramp: \(F_N - F_g\cos{\theta} = 0\) Substituting the weight (\(F_g = Mg\)), the equations become: 1. \(T - Mg\sin{\theta} = 0\) 2. \(F_N - Mg\cos{\theta} = 0\)

4. Solving for Tension

From equation 1, we can solve for tension T: \(T = Mg\sin{\theta}\) Plugging in the given values for mass M, angle θ, and gravitational acceleration (g ≈ 9.81 \(m/s^2\)), we can solve for T: \(T = (64\,\mathrm{kg})(9.81\,\mathrm{m/s^2})\sin(26^\circ)\) Now, calculating the tension, T ≈ 278.22 N So, the tension in the rope is approximately 278.22 Newtons.

Key concepts

Tension in a Rope

Tension is a force that is transmitted through a rope or a string when it is pulled tight by forces acting from opposite ends. Tension is a pulling force since ropes cannot push effectively; they are designed to operate under tension. In our ice cube example, the tension in the rope counteracts the component of the gravitational force pulling the block down the incline.

The tension in the rope must be equal in magnitude to the gravitational force component along the incline to maintain static equilibrium. However, it is important to note that if the rope were not attached, the ice block would slide down due to the unopposed force of gravity. The magnitude of the tension can be calculated using the equilibrium conditions and understanding that the tension force acts along the length of the rope, away from the object it is attached to.

Components of Gravitational Force

Every object with mass experiences a gravitational force which acts towards the center of the planet it is on. In many physics problems, the effects of gravity need to be analyzed as they contribute to the net force acting on an object. For an object on an incline, like our cubical block of ice on the frictionless ramp, the gravitational force can be resolved into two components:

• A component perpendicular to the ramp surface (\(F_g \times \text{cos}(\theta)\))
• A component parallel to the ramp surface (\(F_g \times \text{sin}(\theta)\))

These components of the gravitational force are critical in calculating the normal force exerted by the ramp on the block, as well as any tension in a rope that's holding it in place.

Physics of Forces

In the realm of physics, forces are understood as interactions that change the motion of an object. When multiple forces act on an object, we analyze them using vector representation because forces have both magnitude and direction. The combination of these forces decides the object's net acceleration. In static equilibrium problems, like our block on a ramp, there is no net force acting on the object because all the forces cancel each other out, resulting in no acceleration.

Forces can be broadly categorized into contact forces (such as tension, friction, and normal forces) and non-contact forces (such as gravity). Different forces interact according to Newton's laws of motion, which form the basis for solving problems in classical mechanics.

Equilibrium Equations

Physical systems are in equilibrium when the sum of forces, and often the sum of moments (torques), acting on them is zero, resulting in no change in motion. For static equilibrium, this means an object remains at rest if it was initially at rest. There are two main conditions for equilibrium:

• The sum of all horizontal forces must be zero.
• The sum of all vertical forces must be zero.

In our example, the block of ice is stationary, which tells us that the net force along and perpendicular to the ramp must be zero. By setting up the appropriate equations of equilibrium, as demonstrated in the solution steps, we can solve for unknown quantities like the tension in the rope holding the ice block in place.