You are riding your bike and the tire has a radius of 0.33 m and mass 1.5 kg rotating at 98.7 rad/s. You just notice you are just seconds to riding off a cliff what torque will you need to stop the tire in 2.0 s before you find yourself at the bottom of a 1000 m cliff. New sport extreme mountain bike riding.
Well, that's one way to add a bit of excitement to your bike ride, isn't it? Now, let's see what kind of torque you'll need to prevent yourself from becoming a part of the "extreme mountain bike falling" sport.
To calculate the torque required, we'll need to use the formula:
Torque = Moment of Inertia × Angular Acceleration
Given that the moment of inertia of a solid cylinder is given by:
Moment of Inertia (I) = 0.5 × Mass × Radius^2
We can calculate the moment of inertia:
I = 0.5 × 1.5 kg × (0.33 m)^2
Now, let's find the angular acceleration. We know that the angular acceleration is the change in angular velocity divided by the change in time. You want to stop the tire in 2.0 seconds and the initial angular velocity is 98.7 rad/s. So, the angular acceleration is:
Angular Acceleration = (0 rad/s - 98.7 rad/s) / 2.0 s
Now that we have both the moment of inertia and the angular acceleration, we can calculate the torque:
Torque = (0.5 × 1.5 kg × (0.33 m)^2) × ((0 rad/s - 98.7 rad/s) / 2.0 s)
And there you have it! The torque you'll need to stop the tire in 2.0 seconds before you fall off the cliff. Let's hope you have a good grip on those handlebars! Stay safe, my friend.
To determine the torque needed to stop the tire in 2.0 seconds before reaching the bottom of the 1000 m cliff, we need to calculate the angular acceleration and then use it to find the torque.
Step 1: Calculate the final angular velocity:
The final angular velocity of the tire is 0 rad/s (since we want to stop it).
Step 2: Calculate the angular acceleration:
The angular acceleration can be found using the equation:
angular acceleration (α) = Change in angular velocity (Δω) / Time taken (t)
The change in angular velocity (Δω) is the final angular velocity (0 rad/s) minus the initial angular velocity (98.7 rad/s).
Δω = 0 rad/s - 98.7 rad/s = -98.7 rad/s
The time taken is 2.0 s.
Therefore, the angular acceleration is:
α = Δω / t = -98.7 rad/s / 2.0 s = -49.35 rad/s²
Step 3: Calculate the moment of inertia of the tire:
The moment of inertia (I) of a solid cylinder (such as a bike tire) can be calculated using the equation:
I = (1/2) * mass * radius²
Given that the mass of the tire is 1.5 kg and the radius is 0.33 m:
I = (1/2) * 1.5 kg * (0.33 m)² = 0.272 kg·m²
Step 4: Calculate the torque:
The torque (τ) can be found using the equation:
τ = moment of inertia (I) * angular acceleration (α)
Substituting the values:
τ = 0.272 kg·m² * (-49.35 rad/s²)
τ = -13.428 kg·m²/s²
Therefore, the torque needed to stop the tire in 2.0 seconds before reaching the bottom of the 1000 m cliff is approximately -13.428 kg·m²/s². Note that the negative sign indicates that the torque should be applied in the opposite direction of the initial rotation.
To determine the torque required to stop the tire in 2.0 seconds, we first need to calculate the angular acceleration of the tire.
We can use the formula for angular acceleration:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time
Given:
Initial angular velocity (ω0) = 98.7 rad/s
Final angular velocity (ωf) = 0 rad/s (to stop the tire)
Time (t) = 2.0 seconds
Plugging these values into the formula:
α = (0 - 98.7) / 2.0
= -98.7 / 2.0
= -49.35 rad/s^2
Now that we have the angular acceleration, we can calculate the torque required to stop the tire using the formula:
torque (τ) = moment of inertia (I) * angular acceleration (α)
The moment of inertia of a solid disk rotating about its axis is given by the formula:
I = (1/2) * m * r^2
Given:
Radius (r) = 0.33 m
Mass of the tire (m) = 1.5 kg
Plugging these values into the moment of inertia formula:
I = (1/2) * (1.5 kg) * (0.33 m)^2
= (1/2) * (1.5 kg) * (0.1089 m^2)
= 0.081675 kg*m^2
Now we can calculate the torque:
τ = (0.081675 kg*m^2) * (-49.35 rad/s^2)
= -4.02626 N*m
Therefore, you will need a torque of approximately -4.03 N*m (taking into account the negative sign) to stop the tire in 2.0 seconds before reaching the bottom of a 1000 m cliff.
Hmmm. It is easy to calculate the torque needed to stop the wheel in two seconds. Now it seems to be more important to me to stop the bike before it gets to the cliff, which is quite another calculation.
rotationalimmpule=changeinangularMomentum
Torque*time=momnetInertia*changeangularvel
Torque*2=m*r^2*98.7
solve for torque.
Again, this stops the wheel from rotating in 2 seconds...it has little to do with stopping the bike
the wheel might be stopping on the way down, or the wheel might be stopped at the top, but the bike slides over the cliff, due to insufficient frictional force.