Calculate the angular acceleration of a system at t = 4 seconds using a provided angular velocity graph | Step-by-Step Solution
Problem
Angular velocity of a system: Part A | Problem The figure below shows the angular velocity for a system graphed as a function where all horizontal divisions are in seconds and all vertical divisions are in rad/s. At t = 4 s, the system is at an angular position of -1 rad. What is the angular acceleration (in rad/s²) of the system at t = 4 s?
🎯 What You'll Learn
- Analyze angular velocity graphs
- Calculate angular acceleration from a graph
- Understand relationships between position, velocity, and acceleration
Prerequisites: Basic graphing skills, Understanding of velocity and acceleration, Interpreting graphical representations
💡 Quick Summary
Hi there! I can see you're working with angular kinematics, specifically finding angular acceleration from a graph - this is a great way to visualize rotational motion concepts. The key insight here is thinking about the relationship between angular velocity and angular acceleration - what does angular acceleration represent in terms of how angular velocity changes over time? Take a look at your angular velocity vs. time graph and consider what you'd need to examine at the specific moment when t = 4 seconds. Remember that in kinematics, acceleration is always related to how velocity changes, so think about what mathematical operation you'd use to find the rate of change from a graph. You've got the right setup here, and once you identify what to look for on that graph at t = 4s, the solution will follow naturally!
Step-by-Step Explanation
Hello! Great question about angular kinematics!
What We're Solving:
We need to find the angular acceleration of a system at t = 4 seconds using a graph that shows angular velocity (ω) versus time.The Approach:
Angular acceleration is the rate of change of angular velocity. Just like how linear acceleration is the slope of a velocity-time graph, angular acceleration is the slope of an angular velocity-time graph!The mathematical relationship is: α = dω/dt (angular acceleration = change in angular velocity / change in time)
So we need to find the slope of the ω vs. t graph at t = 4 seconds.
Step-by-Step Solution:
Step 1: Understand what we're looking for
- Angular acceleration (α) has units of rad/s²
- We need the instantaneous angular acceleration at exactly t = 4 s
- Find the point on the curve where t = 4 seconds
- Note the angular velocity value at this point
- If the graph shows a straight line at t = 4 s, calculate: slope = Δω/Δt
- Choose two nearby points and find: (ω₂ - ω₁)/(t₂ - t₁)
- If it's a curve, you'll need the instantaneous slope (tangent line at that point)
- Positive slope = positive angular acceleration
- Negative slope = negative angular acceleration
- Zero slope = zero angular acceleration
The Answer:
Without seeing your specific graph, I can't give you the numerical answer, but once you find the slope of the ω vs. t graph at t = 4 seconds, that value (in rad/s²) will be your angular acceleration!For example, if the slope at t = 4 s is +2 rad/s per second, then α = +2 rad/s².
Memory Tip:
Remember the relationship chain: Position → Velocity → Acceleration- Angular position (θ) → Angular velocity (ω) → Angular acceleration (α)
- Each step to the right involves taking a derivative (finding the slope)!
- ω = dθ/dt and α = dω/dt
⚠️ Common Mistakes to Avoid
- Misreading the graph scales
- Confusing velocity with acceleration
- Not carefully examining the graph's slope changes
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
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