Problem 6 - College Physics

Question

IE oo A hockey puck with a weight of \(0.50 \mathrm{lb}\) is sliding freely across a section of very smooth (frictionless) horizontal ice. (a) When it is sliding freely, how does the upward force of the ice on the puck (the normal force) compare with the upward force when the puck is sitting permanently at rest: (1) The upward force is greater when the puck is sliding; (2) the upward force is less when it is sliding; (3) the upward force is the same in both situations? (b) Calculate the upward force on the puck in both situations.

Step-by-Step Solution

Answer

(a) (3) The upward force is the same in both situations. (b) The upward force is 0.50 lb in both cases.

Step 1: Analyzing the Normal Force in Rest and Motion

First, understand that the normal force is the perpendicular contact force exerted by a surface that opposes the weight of the object. When the puck is at rest, the only forces acting on it are its weight and the normal force. Similarly, when the puck is sliding on a frictionless surface, the same two forces act vertically. Frictionless means no horizontal forces affect the normal force. Thus, in both cases, the normal force is balanced with the weight of the puck.

Step 2: Calculating the Normal Force at Rest

When the puck is at rest, the normal force equals the weight of the puck. Since the puck weighs 0.50 lb, and we know force due to gravity is balanced by the upward normal force, the normal force can be calculated as: \( F_N = 0.50 \text{ lb} \).

Step 3: Calculating the Normal Force in Motion

In the absence of friction and other vertical forces while sliding, the only forces acting on the puck vertically are its weight and the normal force. As established, this situation doesn't affect the normal force. Therefore, the normal force is still equal to the puck's weight: \( F_N = 0.50 \text{ lb} \).

Step 4: Conclusion on Comparison of Normal Force

The normal force when the puck is sliding is the same as when it is at rest because both scenarios involve balancing the puck's weight against the normal force without additional vertical influences.

Key Concepts

Newton's Laws of Motion

Newton's Laws of Motion are fundamental principles that describe how objects move and interact with the forces acting upon them. These laws are crucial for understanding the motion of the hockey puck on the ice.

• First Law (Law of Inertia): An object at rest will stay at rest, and an object in motion will stay in motion at a constant velocity, unless acted upon by an external force. This means that the hockey puck will not accelerate unless a force is applied.
• Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. It can be mathematically expressed as \( F = m \cdot a \), where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration. This law helps determine how much force is needed to move the puck or bring it to a stop on the ice.
• Third Law: For every action, there is an equal and opposite reaction. When the puck exerts a downward force due to its weight, the ice exerts an equal upward force, known as the normal force, balancing the forces vertically.

Understanding these laws helps explain why the puck continues to slide in a straight line at a steady speed when pushed and why no additional force affects its normal force when the surface is frictionless.

Frictionless Surface

A frictionless surface is an ideal concept where no frictional force opposes the movement of an object. In reality, such a perfect surface doesn't exist, but in physics problems, it helps simplify calculations and understand concepts.

On a frictionless surface, like the idealized ice the hockey puck slides upon, horizontal motion remains unaffected by opposing forces. This implies that once the puck is set in motion, it would continue moving indefinitely in the same direction unless acted upon by an external force due to the absence of friction.

This absence of horizontal friction also means the normal force, which acts perpendicularly to the surface to counteract the object's weight, remains constant and unchanged whether the puck is moving or stationary.

• Importance for Learning: Discussing a frictionless surface allows students to focus on vertical forces, illustrating force equilibrium without the complexity of accounting for friction.

Force Equilibrium

Force equilibrium occurs when all the forces acting on an object are balanced, meaning the object is in a state of rest or moves at a constant velocity.

In the context of the hockey puck on ice, equilibrium involves the normal force balancing out the gravitational force exerted by the puck's weight when it is both stationary and during motion. Since the surface is frictionless, the only forces to consider vertically are the weight of the puck and the normal force from the ice.

• Mathematical Representation: In equilibrium, the sum of forces equals zero: \( \sum F = 0 \). Therefore, \( F_N = m \cdot g \), where \( F_N \) represents the normal force, \( m \) the mass of the puck, and \( g \) the acceleration due to gravity.

It is this balance of forces that ensures the normal force remains equal to the weight of the puck in either scenario, highlighting the concept of equilibrium.

Weight and Mass

Weight and mass are related but distinct concepts crucial for understanding forces.

• Mass: It is a measure of the amount of matter in an object and remains constant regardless of location or surface. It is typically measured in kilograms or pounds.
• Weight: It is the force exerted by gravity on an object and can be calculated as the product of mass and gravitational acceleration \( g \). The weight of the hockey puck is given as 0.50 lb.

To calculate the weight from mass, the formula \( W = m \cdot g \) is used, where \( W \) is weight, \( m \) is mass, and \( g \) is gravitational acceleration (approximately 9.8 m/s² on Earth).

Understanding the difference between weight and mass is essential in physics problems, as it helps in accurately identifying forces acting on objects, like the puck, and ensures comprehension of how these forces contribute to equilibrium and motion.