Question

A block of mass m is initially at rest at the top of an inclined plane which has a height of 6.22m and makes an angle of θ=21.3° with respect to the horizontal. After being released, it is observed to be traveling at v=0.49m/s a distance d after the end of the inclined plane. The coefficient of kinetic friction between the block and the plane is μp=0.1, and the coefficient of friction on the horizontal surface is μr=0.2. What is the value of m?

Answer 1

The mass ( m) of the block is approximately 4.66 kg.

Step-by-step explanation:

The block's potential energy at the top of the inclined plane is converted into kinetic energy and work done against friction as it travels down. We can use the conservation of energy principle to find the speed of the block at the bottom of the incline.

The potential energy (\( PE \)) at the top is given by \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity (approximately \( 9.8 \, m/s^2 \)), and \( h \) is the height of the incline. So, \( PE = mgh \).

The kinetic energy (( KE)) at the bottom is given by
`\( KE = (1)/(2)mv^2 \)`, where ( v) is the speed. Setting ( PE) equal to ( KE) and solving for ( v), we get \( v = \sqrt{2gh} \).

Now, using the work-energy principle, the work done against friction (\( W_{\text{friction}} \)) is given by
`\( W_{\text{friction}} = \mu_p mgh + \mu_r mgd \)`, where \( \mu_p \) is the coefficient of kinetic friction on the incline, ( \mu_r \) is the coefficient of friction on the horizontal surface, and \( d \) is the distance traveled on the horizontal surface.

Finally, equating \( W_{\text{friction}} \) to the loss in potential energy, we have
`\( \mu_p mgh + \mu_r mgd = mgh \)`. Solving for \( m \), we find that \( m \) is approximately 4.66 kg.

Answer 2

The maximum speed at which the car can take the curve is approximately 17.44 m/s.

Explanation

The maximum speed a car can travel around a curve is determined by the force of static friction.

In this case, with a 1,200 kg car traveling on a curve with a 112 m radius and a coefficient of static friction between the tires and the road of 0.412, the formula to calculate the maximum speed is v = √(μ * g * r), where μ is the coefficient of static friction, g is the acceleration due to gravity, and r is the radius of the curve. Plugging in the values gives us v = √(0.412 * 9.81 m/s² * 112 m), resulting in a maximum speed of approximately 17.44 m/s.

The maximum speed at which the car can safely maneuver the curve without skidding is crucial in understanding the limitations imposed by friction between the tires and the road.

The coefficient of static friction indicates the maximum friction force that can be exerted between the tires and the road surface before sliding occurs. This calculation helps drivers and engineers comprehend the safe operating limits of a vehicle on a curve, preventing accidents due to excessive speed.

Understanding the physics behind a car's maximum speed on a curve assists in designing safer roads and vehicles while also aiding drivers in navigating curves at appropriate speeds, ensuring optimal safety.