Can you solve THIS from the British Physics Olympiad?
TLDRIn this engaging physics problem from the British Olympiad, Emmy uses bathroom scales in a lift to measure the acceleration of the lift as a function of time. By applying Newton's second law and integrating the acceleration, she calculates the speed after 10 seconds and the remaining distance to the 30th floor. Finally, she determines the minimum mass reading on the scales during deceleration, providing a comprehensive insight into the dynamics of the lift's motion.
Takeaways
- 🔍 Emmy uses bathroom scales in a lift to measure the lift's acceleration by observing the scale's reading as a function of time.
- 📚 Newton's second law is applied to find the relationship between the normal force, weight, and resultant acceleration in the lift.
- 🧭 The perceived weight on the scales is equal to the normal reaction force divided by gravity (g), leading to an expression for the measured mass.
- 📉 The mass at time t=0 is given as 60 kg, which is the person's actual mass when the lift is not moving.
- ⚖️ By equating the expressions for the measured mass and the actual mass, an equation for the lift's acceleration as a function of time is derived.
- ⏱ The lift's speed after 10 seconds is calculated by integrating the acceleration with respect to time, resulting in a quadratic expression.
- 📈 The constant of integration is determined to be zero because the lift is not moving at t=0, leading to a specific expression for the speed at 10 seconds.
- 🏢 The lift's displacement after 10 seconds is found by integrating the speed, which is used to calculate the remaining distance to the 30th floor.
- 🛑 After the initial 10 seconds, the lift decelerates at a constant rate to reach the 30th floor, requiring the use of kinematic equations to find the minimum acceleration.
- 📊 The minimum mass reading on the scales during deceleration is calculated by considering the minimum acceleration and the normal force equation.
- 🔚 The final perceived mass reading on the scales, during the deceleration phase, is found to be 15 kg, which is the minimum reading due to the deceleration.
Q & A
What is the equation used to describe the variation in the reading on the scales with time?
-The reading on the scales varies with time according to the equation: 1 + (t/10) - (t^2/100), where t is time.
How is Newton's Second Law applied to analyze the forces in the lift?
-Newton's Second Law is applied by considering the balance of forces in the lift. The normal force N acting upwards and the weight mg acting downwards lead to the equation: N - mg = ma, where a is the acceleration of the lift.
What is the expression for the perceived weight of Emmy as a function of time?
-The perceived weight, which corresponds to the normal reaction force divided by g, is given by: m(a + g)/g. This can be simplified to: m(1 + (t/10) - (t^2/100)) where t is time.
How is the acceleration of the lift derived as a function of time?
-The acceleration a(t) of the lift as a function of time is derived by equating the perceived weight to the measured mass and simplifying. The resulting expression is: a(t) = g((t/10) - (t^2/100)).
What integral is used to find the velocity of the lift after 10 seconds?
-The velocity after 10 seconds is found by integrating the acceleration function: v(t) = ∫a(t) dt. Specifically, v(t) = g((t^2/20) - (t^3/300)), with the constant of integration being zero.
What is the velocity of the lift after 10 seconds?
-The velocity of the lift after 10 seconds is calculated to be 16.35 meters per second.
How is the distance traveled by the lift in the first 10 seconds calculated?
-The distance traveled in the first 10 seconds is calculated by integrating the velocity function: s(t) = ∫v(t) dt. The expression obtained is s(t) = g((t^3/60) - (t^4/1200)), which results in a distance of 81.75 meters after 10 seconds.
How is the deceleration of the lift calculated when it travels the remaining distance?
-The deceleration is calculated using the equation: a = (v^2 - u^2)/(2s), where u is the initial velocity (16.35 m/s), v is the final velocity (0 m/s), and s is the remaining distance (18.25 meters). The deceleration is found to be -7.3 m/s².
What is the minimum reading on the scales when the lift decelerates?
-The minimum reading on the scales is calculated using the equation: N = m(a + g)/g, where a is the deceleration (-7.3 m/s²). The minimum reading is found to be 15 kg.
What physical principles are primarily used in solving this problem?
-The problem primarily utilizes Newton's Second Law of Motion, integration for finding velocity and displacement, and kinematic equations to analyze the motion of the lift.
Outlines
🔍 Lift Motion Analysis
The script begins with a physics problem from the British Olympiad involving Emmy in a lift with a set of scales. It describes the process of determining the acceleration of the lift based on the perceived weight change over time. Emmy uses Newton's second law to relate the normal force, weight, and mass of the person to the resultant acceleration. The perceived weight is equated to the normal reaction force divided by gravity, leading to an expression for the measured mass. Given the mass at time zero is 60 kg, the script derives an equation for the acceleration of the lift as a function of time. The problem then moves to calculating the speed of the lift after 10 seconds by integrating the acceleration with respect to time, resulting in a speed of 16.35 meters per second. The script concludes with the calculation of the minimum mass reading during deceleration, which is found to be 15 kg.
📏 Calculating Displacement and Deceleration
This paragraph continues the physics problem by calculating the displacement of the lift during the initial 10 seconds of motion, using the previously derived speed equation. The displacement is found by integrating the speed equation, resulting in a distance of 81.75 meters. Since the building's height is 100 meters, the remaining distance to be covered is 18.25 meters, which will be done at a constant deceleration rate. The script then uses the SUVAT equations (a set of equations relating displacement, initial velocity, final velocity, acceleration, and time) to determine the minimum acceleration during deceleration, which is -7.3 m/s². Finally, the script calculates the minimum reading on the scales during deceleration, which corresponds to the minimum acceleration experienced, resulting in a perceived mass of 15 kg.
Mindmap
Keywords
💡Lift
💡Bathroom Scales
💡Newton's Second Law
💡Normal Force
💡Perceived Weight
💡Acceleration
💡Integration
💡Velocity
💡Displacement
💡Deceleration
💡Mass Reading
Highlights
Emmy uses bathroom scales in a lift to measure acceleration.
Newton's second law is applied to the balance of forces in the lift.
The normal reaction is equal to mass times acceleration plus gravitational force.
The perceived weight equals the normal reaction divided by gravity.
The mass of the person is determined to be 60 kg at time t equals zero.
An equation is derived for the acceleration of the lift as a function of time.
Integration of acceleration over time gives the velocity of the lift.
The speed of the lift after 10 seconds is calculated to be 16.35 m/s.
The lift's displacement during the initial 10 seconds is calculated.
The remaining distance to the 30th floor is 18.25 meters after the initial acceleration.
Suvard's equations are used to calculate deceleration.
The minimum acceleration during deceleration is found to be -7.3 m/s².
The minimum mass reading on the scales during deceleration is calculated.
The perceived mass reading is 15 kg at the minimum acceleration.
The video encourages viewers to explore more problems from the physics olympiad.
Transcripts
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