Problem 9 - University Physics with Modern Physics

Question

A man pushes on a piano with mass 180 \(\mathrm{kg}\) so that it slides at constant velocity down a ramp that is inclined at \(11.0^{\circ}\) above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes (a) parallel to the incline and (b) parallel to the floor.

Step-by-Step Solution

Answer

(a) 342 N parallel to the incline; (b) 348 N parallel to the floor.

Step 1: Identify Given Information

The mass of the piano is given as 180 kg and the angle of the incline is 11.0 degrees. The problem states that the piano slides at constant velocity, which implies that there is no net force acting on the piano.

Step 2: Calculate Gravitational Force Component Parallel to Incline (a)

When an object is on an inclined plane, the gravitational force acting on it can be decomposed into two components: one parallel to the incline and one perpendicular to the incline. The force component parallel to the incline is given by \( F_{gx} = m \cdot g \cdot \sin(\theta) \), where \( m = 180 \) kg, \( g = 9.8 \) m/s², and \( \theta = 11^{\circ} \). Substituting these values gives \( F_{gx} = 180 \times 9.8 \times \sin(11^{\circ}) \approx 342 \) N.

Step 3: Apply Condition for Constant Velocity (a)

For the piano to slide at constant velocity down the incline, the force applied by the man, \( F_m \), must be equal in magnitude to the gravitational component parallel to the incline. Therefore, the force applied by the man is \( F_m = 342 \) N, parallel to the incline.

Step 4: Calculate Gravitational Force Component Parallel to Floor (b)

When considering the force applied parallel to the floor, the man has to counteract the entire weight component that would cause the piano to slide down the ramp. The horizontal component of the applied force can be calculated using the relationship \( F_{mf} = F_{gx} / \cos(\theta) \). Using \( F_{gx} = 342 \) N and \( \theta = 11^{\circ} \), we find \( F_{mf} \approx 342 / \cos(11^{\circ}) \approx 348 \) N.

Step 5: Apply Condition for Constant Velocity (b)

For the scenario where the man pushes parallel to the floor, the force applied by him should counteract the gravitational pull along the incline. Therefore, the required force magnitude when pushing parallel to the floor is about \( 348 \) N.

Key Concepts

Gravitational Force Components

When dealing with objects on an incline, understanding gravitational force components is vital. These components help us analyze how gravity affects the object in different directions.

Gravity, a force that pulls objects towards Earth, acts straight down. However, when an object like a piano is placed on an inclined plane, this gravitational force can be divided into two simpler components:

• Parallel to the incline - helps us determine how much force is needed to make the object slide up or down the ramp.
• Perpendicular to the incline - makes it easier to understand the normal force (the force usually acting opposite to weight).

In our exercise, we focus on the parallel component because it tells us how much force the man must apply to keep the piano moving at a constant speed. The formula used is: \( F_{gx} = m \cdot g \cdot \sin(\theta) \), where \( m \) is mass, \( g \) is acceleration due to gravity, and \( \theta \) is the incline angle. By plugging in the specific values, we find this force to be 342 N.

This understanding is crucial because it's the amount of force needed to overcome gravity and keep the piano sliding steadily down the ramp.

Inclined Plane Dynamics

Inclined plane dynamics looks at how forces act on objects that are on a slope. This setup often makes physics problems a bit more complex because angles are involved, changing the direction and magnitude of forces.

Let's break it down:

• The incline itself changes how forces like gravity and any applied force affect the object.
• Objects on inclines need enough force exerted either down or up the plane for them to stay steady or move smoothly.

For our piano example, when the man is pushing it, he's applying a force against the gravitational pull on the ramp. This means he's essentially balancing the force to move the piano steadily.

When acting parallel to the incline, the man must apply a force exactly equal to the gravitational force component parallel to the incline to maintain constant velocity, meaning the object isn't speeding up or slowing down. If he pushes parallel to the floor, such application involves additional calculations to account for the inclined angle, effectively needing more force due to the angle’s geometry. This approach calculates the required force using the formula \( F_{mf} = F_{gx} / \cos(\theta) \), resulting in 348 N needed for the piano to maintain its motion.

Constant Velocity Condition

The constant velocity condition simplifies problem-solving enormously because it means that forces are balanced perfectly. When an object moves at constant speed, its net force equals zero under Newton’s First Law. This condition is the key to solving our piano problem. The piano slides down the ramp without speeding up or slowing down.

Here's how it works:

• At constant velocity, the force the man applies either parallel to the incline or the floor must equal the gravitational force component.
• It indicates that the forces acting on the piano are in equilibrium, so the sum of forces results in no acceleration, just uniform movement.

In practical terms for our exercise, a 342 N force applied along the incline or a slightly greater 348 N parallel to the floor force ensures the piano’s constant slide down the ramp.

This highlights the principle that different directions of force application require altered calculations to achieve the same non-accelerating motion. Understanding this condition helps to navigate through many physics problems involving constant motions.