Problem 89 - College Physics

Question

If the toboggan is well waxed so that there is no friction, what force does the sail have to provide to move the student-filled toboggan up the hill at a constant velocity (once it gets started)? \(\begin{array}{l}{\text { A. } g \sin \alpha} \\ {\text { B. } w \sin \alpha} \\\ {\text { C. } w} \\ {\text { D. } w g \sin \alpha}\end{array}\)

Step-by-Step Solution

Answer

Option B: \( w \sin \alpha \).

Step 1: Understanding the Problem

The problem asks for the force required to move a student-filled toboggan up a hill at a constant velocity. This means the net force parallel to the hill must be zero as there is no acceleration. The force provided by the sail must counteract the component of gravitational force pulling the toboggan down the slope.

Step 2: Identifying the Forces

Since there is no friction, the only force working against the toboggan is gravity. This force can be broken down into two components: one parallel to the hill and one perpendicular. We need to find the parallel component.

Step 3: Resolving Gravitational Force

The gravitational force acting on the toboggan has a parallel component given by \( F_{gravity} = w \sin \alpha \), where \( w \) is the weight of the toboggan (i.e., the gravitational force vertically downward) and \( \alpha \) is the angle of the hill.

Step 4: Finding the Sail Force

To move the toboggan up the hill at a constant velocity, the sail must exert a force equal in magnitude and opposite in direction to the parallel component of the gravitational force. Therefore, the sail force \( F_{sail} = w \sin \alpha \).

Step 5: Selecting the Correct Option

The correct option corresponds to \( w \sin \alpha \), which is option B.

Key Concepts

Gravitational Force Component

When dealing with physics problems involving motion on an inclined surface like a hill, understanding gravitational force components is crucial. Gravity pulls objects straight downwards towards the center of Earth. However, when an object is on a slope, this gravitational force can be split into two components:

• The component parallel to the slope, which tries to pull the object down the hill.
• The component perpendicular to the slope, which presses the object against the hill.

In our specific scenario, we are interested in the parallel component because it affects the toboggan's uphill motion. We calculate this parallel component using the formula: \( F_{gravity} = w \sin \alpha \), where \( w \) is the weight of the toboggan and \( \alpha \) is the angle of the hill. This component determines how much force is needed to counteract gravity along the hill's incline.

Toboggan Motion

Tobogganing uphill might seem counterintuitive because we usually associate toboggans with downhill racing. But when a student-filled toboggan moves uphill, understanding forces becomes even more interesting.
When the toboggan is on a hill, gravity acts against the intended uphill motion by pulling it downward. If there is no friction, as stated in the problem, the only opposition to moving uphill comes from this gravitational force component. The sail provides an uphill force to counter this effect, maintaining motion at a stable rate. It's important to remember that in this situation, friction is not considered, simplifying our analysis to essentially battling gravity.

Constant Velocity

A key concept in this exercise is the idea of constant velocity. Constant velocity implies that the speed and direction of the toboggan’s motion are steady over time.
In physics, when an object moves at constant velocity, the forces acting on it must be balanced. This means the net force on the toboggan along the hill must be zero, as there is no acceleration or deceleration. For our exercise, the force exerted by the sail must exactly match and counteract the gravitational force pulling the toboggan back down the slope. This equilibrium is what keeps the velocity constant during the upward motion.

Force Analysis

Analyzing forces is vital to solving this problem successfully. Without friction, the primary forces we focus on are the gravitational forces and the sail force.Here's how we break it down:

• Gravitational Force: It has both parallel and perpendicular components. Our focus here is the parallel component wanting to bring the toboggan downhill.

• Sail Force: This is the force required to negate the gravitational pull along the slope. Since the toboggan moves at constant velocity, the sail must provide a force equal in magnitude to the parallel gravitational component but in the opposite direction.

This results in the balanced force equation: \( F_{sail} = w \sin \alpha \). Understanding these forces and their interactions aids in predicting and controlling the toboggan's motion on the hill.