A stationary bicycle is raised off the ground, and its front wheel (m = 1.3 kg) is rotating at an angular velocity of 14.0 rad/s (see the drawing). The front brake is then applied for 3.0 s, and the wheel slows down to 2.7 rad/s. Assume that all the mass of the wheel is concentrated in the rim, the radius of which is 0.33 m. The coefficient of kinetic friction between each brake pad and the rim is ìk = 0.85. What is the magnitude of the normal force that each brake pad applies to the rim?
To find the magnitude of the normal force that each brake pad applies to the rim, we can use the equation for torque.
The torque acting on the wheel is given by τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The moment of inertia of a solid cylinder about its central axis is given by I = 1/2 * m * r^2, where m is the mass of the wheel and r is the radius of the wheel.
Given:
Mass of the front wheel, m = 1.3 kg
Radius of the wheel, r = 0.33 m
Angular velocity before braking, ωi = 14.0 rad/s
Angular velocity after braking, ωf = 2.7 rad/s
Time taken for braking, t = 3.0 s
Coefficient of kinetic friction, μk = 0.85
First, let's find the initial angular acceleration:
ωf = ωi + α * t
α = (ωf - ωi) / t
Substituting the given values:
α = (2.7 rad/s - 14.0 rad/s) / 3.0 s
α = -3.7667 rad/s^2 (negative sign indicates deceleration)
Next, let's calculate the moment of inertia:
I = 1/2 * m * r^2
I = 1/2 * 1.3 kg * (0.33 m)^2
I = 0.070785 kg·m^2
Now, let's calculate the torque:
τ = I * α
τ = 0.070785 kg·m^2 * -3.7667 rad/s^2
τ = -0.2666 N·m
The torque exerted by the friction between the brake pads and the rim is equal in magnitude and opposite in direction to the torque exerted by the wheel.
The torque due to friction is given by τ = r * F, where r is the radius of the wheel and F is the force exerted by each brake pad.
Solving for F:
F = τ / r
F = (-0.2666 N·m) / 0.33 m
F = -0.8076 N (negative sign indicates opposite direction)
Since the force due to friction is negative, the direction of the force applied by each brake pad is opposite to the direction of motion.
The normal force, N, is given by N = F / μk, where μk is the coefficient of kinetic friction.
Substituting the given values:
N = (-0.8076 N) / 0.85
N = -0.9496 N (negative sign indicates opposite direction)
So, the magnitude of the normal force that each brake pad applies to the rim is approximately 0.9496 N.
To find the magnitude of the normal force that each brake pad applies to the rim, we can use the equation for torque.
Torque (τ) is defined as the product of the force (F) applied to an object and the distance (r) from the point of rotation. Mathematically, it can be written as τ = F * r.
In this case, the torque due to the frictional force between the brake pad and the rim is what causes the angular deceleration of the wheel.
Given:
m (mass of the wheel) = 1.3 kg
ωi (initial angular velocity) = 14.0 rad/s
ωf (final angular velocity) = 2.7 rad/s
R (radius of the rim) = 0.33 m
μk (coefficient of kinetic friction) = 0.85
t (time) = 3.0 s
To find the torque, we can use the rotational analogue of Newton's second law, which states that τ = I * α, where I is the moment of inertia and α is the angular acceleration.
For a solid disc, the moment of inertia is given by I = (1/2) * m * R^2.
Now, we need to find the angular acceleration (α). We can use the equation α = (ωf - ωi) / t.
Substituting the given values:
α = (2.7 - 14.0) / 3.0 = -11.3 / 3.0 = -3.77 rad/s^2 (negative because the wheel is decelerating)
The torque can now be calculated using τ = I * α:
τ = (1/2) * 1.3 * (0.33^2) * -3.77
Next, we need to find the frictional force (F) using the equation τ = F * r:
F = τ / r
Finally, the normal force (N) is equal to the frictional force (F) because the vertical component of the normal force balances the weight of the wheel.
Therefore, the magnitude of the normal force that each brake pad applies to the rim is N = F.
Calculate F using the derived values and substitute it into the equation.
Finally, calculate N using the calculated F.
Note: Please let me know if any conversion of units is required, such as converting the radius from meters to centimeters.
Torque = (moment of inertia)*(angular deceleration rate)
Solve for torque and relate that to the brake pads force.
Torque = 2*(normal force)*0.85*R