Problem 16 - College Physics

Question

\(\bullet$$\bullet\) A man pushes on a piano of mass 180 \(\mathrm{kg}\) so that it slides at a constant velocity of 12.0 \(\mathrm{cm} / \mathrm{s}\) down a ramp that is inclined at \(11.0^{\circ}\) above the horizontal. No appreciable friction is acting on the piano. Calculate the magnitude and direction of this push (a) if the man pushes parallel to the incline, (b) if the man pushes the piano up the plane instead, also at 12.0 \(\mathrm{cm} / \mathrm{s}\) paral- lel to the incline, and (c) if the man pushes horizontally, but still with a speed of 12.0 \(\mathrm{cm} / \mathrm{s}\) .

Step-by-Step Solution

Answer

(a) 337 N downward along the plane, (b) 674 N upward, (c) 342 N horizontally.

Step 1: Analyze the Forces Involved

The forces acting on the piano include the gravitational force, the normal force from the ramp, and the force applied by the man. Since there's no appreciable friction, we don't consider a friction force.

Step 2: Determine Gravitational Force Components

First, calculate the gravitational force acting down the ramp. The gravitational force can be decomposed into two components relative to the inclined plane: a component parallel to the plane and a component perpendicular to it. The weight of the piano is given by: \[ F_g = m \cdot g = 180 \times 9.8 = 1764 \text{ N} \]The component of gravitational force parallel to the incline is: \[ F_{g,\parallel} = F_g \cdot \sin(\theta) = 1764 \cdot \sin(11.0^{\circ}) \approx 337 \text{ N} \]

Step 3: Calculate Required Force for Parallel Push (a)

To move the piano at constant velocity down the ramp, the man must exert a force of the same magnitude but opposite in direction to the gravitational component parallel to the plane. Thus, if the man pushes downward along the plane, the force is:\[ F_{\text{man}} = F_{g,\parallel} = 337 \text{ N} \]

Step 4: Calculate Force for Upward Push (b)

If the man pushes the piano up the incline, the applied force must overcome both the gravitational component down the incline and the opposite motion at constant velocity:\[ F_{\text{man}} = 2 \times F_{g,\parallel} = 2 \times 337 = 674 \text{ N} \]

Step 5: Calculate Horizontal Push Force (c)

For a horizontal push, we must consider the angle of the ramp. The horizontal force required can be calculated using trigonometric identities:The force exerted horizontally \( F_{h} \) creates a component along the incline that matches the gravitational force:\[ F_{h} \cdot \cos(11.0^{\circ}) = F_{g,\parallel} \]Solving for \( F_{h} \):\[ F_{h} = \frac{337}{\cos(11.0^{\circ})} \approx 342 \text{ N} \]

Step 6: Summarize and Validate

Summarizing the solutions: - For (a) parallel push: 337 N downward along plane. - For (b) upward push: 674 N upward along plane. - For (c) horizontal push: 342 N horizontally. Each calculation matches the requirement to maintain constant velocity, validating the approach.

Key Concepts

Inclined Plane

Inclined planes are flat surfaces tilted at an angle. They help us move objects to different heights more easily. When you place an object on an inclined plane, gravity pulls it down along the slope and presses it against the surface.
Key things to know about inclined planes:

• Gravity acts on the object both parallel and perpendicular to the plane.
• The angle of the incline affects how strong these gravitational components are.
• Without friction, objects will slide down more easily.

In the given problem, the inclined plane is at an angle of 11° to the horizontal. The piano slides down without friction, simplifying calculations since you don't have to account for frictional forces.

Newton's Laws of Motion

Newton's Laws of Motion are fundamental in understanding how objects behave under forces.

• First Law: An object in motion stays in motion unless acted on by an external force.
• Second Law: The acceleration of an object is proportional to the net force acting on it. Formula: \[ F = m \cdot a \]
• Third Law: For every action, there's an equal and opposite reaction.

In this problem, we use Newton's Second Law. Since the piano moves at a constant velocity, the net force along the direction of movement is zero. So, the man's push balances the component of gravitational force parallel to the incline.

Gravitational Force

Gravitational force is the force by which Earth attracts objects towards its center. This force can be expressed mathematically.
The formula for gravitational force is: \[ F_g = m \times g \]Where:

• \( F_g \) is the gravitational force in newtons (N),
• \( m \) is the mass of the object in kilograms (kg),
• \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth).

In this exercise, the gravitational force acting on the piano is 1764 N. This force splits into components, and we focus on the one along the inclined plane for moving the piano.

Vector Components

Vector components break a force into different directions, making complex problems simpler. For an inclined plane, decompose the gravitational force into parallel and perpendicular components.
Mathematically, you find these components using:

• Parallel Component: \( F_{g,\parallel} = F_g \times \sin(\theta) \)
• Perpendicular Component: \( F_{g,\perp} = F_g \times \cos(\theta) \)

In our scenario, because there's no friction, the primary focus is the parallel component. This determines how the piano moves down the slope. Calculating it, we get about 337 N pushing down the incline, forming the core of this problem's solution. It's essential for aligning the man's push to keep the piano moving steadily.