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A four-wheel vehicle of mass 1000 kg moves uniformly in a straight line with the wheels revolving at 10 rad/s. The wheels are identical, each with a radius of 0.2 m. Then a constant braking torque is applied to all the wheels and the vehicle experiences a uniform deceleration. For the vehicle to stop in 10 s, the braking torque (in N.m) on each wheel is _______
(Important - Enter only the numerical value in the answer)
Correct answer is between '9,11'. Can you explain this answer?
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To calculate the braking torque on each wheel of the vehicle, we can use the equation for angular acceleration:
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The moment of inertia for a solid wheel can be calculated using the formula:
I = 0.5 * m * r^2
where m is the mass of the wheel and r is its radius.
Since all the wheels are identical, we can calculate the moment of inertia for one wheel and then multiply it by 4 to get the total moment of inertia for all the wheels.
Given:
Mass of the vehicle (m) = 1000 kg
Radius of the wheel (r) = 0.2 m
Angular velocity (ω) = 10 rad/s
Time taken to stop (t) = 10 s
Step 1: Calculate the moment of inertia for one wheel
Using the formula for moment of inertia, we can calculate the moment of inertia for one wheel:
I = 0.5 * m * r^2
= 0.5 * 1000 kg * (0.2 m)^2
= 0.5 * 1000 kg * 0.04 m^2
= 20 kg.m^2
Step 2: Calculate the angular deceleration
The angular deceleration (α) can be calculated using the formula:
α = (ωf - ωi) / t
where ωf is the final angular velocity (0 rad/s) and ωi is the initial angular velocity (10 rad/s).
α = (0 rad/s - 10 rad/s) / 10 s
= -1 rad/s^2
Step 3: Calculate the braking torque on each wheel
The braking torque can be calculated using the equation:
τ = Iα
τ = (20 kg.m^2) * (-1 rad/s^2)
= -20 N.m
Since torque is a vector quantity, the negative sign indicates that the torque is in the opposite direction of the initial motion. However, for the purpose of calculating the magnitude of the torque, we can ignore the negative sign.
So, the magnitude of the braking torque on each wheel is 20 N.m.
The correct answer is between 9 and 11 N.m, which is not obtained in this calculation. There may be some other considerations or assumptions made in the solution that are not mentioned in the question.
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The moment of inertia for a solid wheel can be calculated using the formula:
I = 0.5 * m * r^2
where m is the mass of the wheel and r is its radius.
Since all the wheels are identical, we can calculate the moment of inertia for one wheel and then multiply it by 4 to get the total moment of inertia for all the wheels.
Given:
Mass of the vehicle (m) = 1000 kg
Radius of the wheel (r) = 0.2 m
Angular velocity (ω) = 10 rad/s
Time taken to stop (t) = 10 s
Step 1: Calculate the moment of inertia for one wheel
Using the formula for moment of inertia, we can calculate the moment of inertia for one wheel:
I = 0.5 * m * r^2
= 0.5 * 1000 kg * (0.2 m)^2
= 0.5 * 1000 kg * 0.04 m^2
= 20 kg.m^2
Step 2: Calculate the angular deceleration
The angular deceleration (α) can be calculated using the formula:
α = (ωf - ωi) / t
where ωf is the final angular velocity (0 rad/s) and ωi is the initial angular velocity (10 rad/s).
α = (0 rad/s - 10 rad/s) / 10 s
= -1 rad/s^2
Step 3: Calculate the braking torque on each wheel
The braking torque can be calculated using the equation:
τ = Iα
τ = (20 kg.m^2) * (-1 rad/s^2)
= -20 N.m
Since torque is a vector quantity, the negative sign indicates that the torque is in the opposite direction of the initial motion. However, for the purpose of calculating the magnitude of the torque, we can ignore the negative sign.
So, the magnitude of the braking torque on each wheel is 20 N.m.
The correct answer is between 9 and 11 N.m, which is not obtained in this calculation. There may be some other considerations or assumptions made in the solution that are not mentioned in the question.
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