Chapter 4: Problem 5
A hockey player hits a puck of mass \(m_{P}\) into a semicircular groove in the ice of radius \(R\). The ice has coefficient of Coulomb friction \(\mu_{c}\). What is the impulse that must be imparted to the puck at the entrance to the groove such that it makes it around the semicircle exactly once before stopping?
Short Answer
Step by step solution
Understanding the Problem
Determine Frictional Force
Calculate Work Done by Friction
Relate Work to Initial Kinetic Energy
Solve for Initial Velocity
Calculate Impulse
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Impulse
The formula for impulse is given by:
- Impulse ( \( J \)) = Change in momentum = Final momentum - Initial momentum
- If an initial velocity ( \( v \)) is imparted to the stationary puck of mass ( \( m_{P} \)), then impulse can also be calculated as:
- \( J = m_{P} \, v \)
Frictional Force
The formula to calculate frictional force (\( F_{f} \)) is:
- \( F_{f} = \mu_{c} \cdot m_{P} \cdot g \)
This force is always directed opposite to the direction of motion, reducing the puck's kinetic energy over time.
Kinetic Energy
The formula for kinetic energy (\( KE \)) of an object is given by:
- \( KE = \frac{1}{2} m_{P} v^2 \)
Coulomb Friction
Mathematically, Coulomb friction is represented as:
- \( F_{f} = \mu_{c} \cdot N = \mu_{c} \cdot m_{P} \cdot g \)
Understanding this principle is vital for calculating the work done by friction and thus the initial impulse required.